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Let \( {S}^{r} \) denote the \( r \) -sphere. Then \( {\pi }_{1}\left( {S}^{1}\right) \cong \mathbb{Z} \), while \( {S}^{r} \) is simply-connected if \( r \geq 2 \) . | We may identify the circle \( {S}^{1} \) with the unit circle in \( \mathbb{C} \) . Then \( x \mapsto {\mathrm{e}}^{2\pi ix} \) is a covering map \( \mathbb{R} \rightarrow {S}^{1} \) . The space \( \mathbb{R} \) is contractible and hence simply-connected, so it is the universal covering space. If we give \( {S}^{1} \subset {\mathbb{C}}^{ \times } \) the group structure it inherits from \( {\mathbb{C}}^{ \times } \), then this map \( \mathbb{R} \rightarrow {S}^{1} \) is a group homomorphism, so by Theorem 13.2 we may identify the kernel \( \mathbb{Z} \) with \( {\pi }_{1}\left( {S}^{1}\right) \) . To see that \( {S}^{r} \) is simply connected for \( r \geq 2 \), let \( p : \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{r} \) be a path. Since it is a mapping from a lower-dimensional manifold, perturbing the path slightly if necessary, we may assume that \( p \) is not surjective. If it omits one point \( P \in {S}^{r} \), its image is contained in \( {S}^{r} - \{ P\} \), which is homeomorphic to \( {\mathbb{R}}^{r} \) and hence contractible. Therefore \( p \), is path-homotopic to a trivial path. | Yes |
Show that there is a set \( A \) of reals of cardinality \( \mathfrak{c} \) such that \( A \cap C \) is countable for every closed, nowhere dense set. | Null | No |
Show that the discriminant is well-defined. In other words, show that given \( {\omega }_{1},{\omega }_{2},\ldots ,{\omega }_{n} \) and \( {\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{n} \), two integral bases for \( K \), we get the same discriminant for \( K \). | Null | No |
Lemma 3.2. \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}\left( x\right) \) exist in \( \mathfrak{A} \) and depends only on \( x \) and \( \Phi \), not on the choice of \( \left\{ {f}_{n}\right\} \) . | Proof of Lemma 3.2. Choose \( \gamma \) as in Exercise 3.2. Then
\[
\begin{Vmatrix}{{f}_{n}\left( x\right) - \frac{1}{2\pi i}{\int }_{\gamma }\frac{\Phi \left( t\right) {dt}}{t - z}}\end{Vmatrix} = \begin{Vmatrix}{\frac{1}{2\pi i}{\int }_{\gamma }\frac{{f}_{n}\left( t\right) - \Phi \left( t\right) }{t - x}{dt}}\end{Vmatrix}
\]
\[
\leq \frac{1}{2\pi }{\int }_{\gamma }\left| {{f}_{n}\left( t\right) - \Phi \left( t\right) }\right| \begin{Vmatrix}{\left( t - x\right) }^{-1}\end{Vmatrix}{dx}
\]
\( \rightarrow 0 \) as \( n \rightarrow \infty \), since \( \begin{Vmatrix}{\left( t - x\right) }^{-1}\end{Vmatrix} \) is bounded on \( \gamma \) while \( {f}_{n} \rightarrow \Phi \) uniformly on \( \gamma \) . Thus
(5)
\[
\mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}\left( x\right) = \frac{1}{2\pi i}{\int }_{\gamma }\frac{\Phi \left( t\right) {dt}}{t - x}.
\] | No |
Exercise 4.4.7. Let \( X \) be a compact metric space, and assume that \( {\nu }_{n} \rightarrow \mu \) in the weak*-topology on \( \mathcal{M}\left( X\right) \) . Show that for a Borel set \( B \) with \( \mu \left( {\partial B}\right) = 0 \) , | Show that for a Borel set \( B \) with \( \mu \left( {\partial B}\right) = 0 \) ,\[
\mathop{\lim }\limits_{{n \rightarrow \infty }}{\nu }_{n}\left( B\right) = \mu \left( B\right)
\] | Yes |
Theorem 14 The list-chromatic index of a bipartite graph equals its chromatic index. | Let \( G \) be a bipartite graph with bipartition \( \left( {{V}_{1},{V}_{2}}\right) \), and let \( \lambda : E\left( G\right) \rightarrow \) \( \left\lbrack k\right\rbrack \) be an edge-colouring of \( G \), where \( k \) is the chromatic index of \( G \) . Define preferences on \( G \) as follows: let \( a \in {V}_{1} \) prefer a neighbour \( A \) to a neighbour \( B \) iff \( \lambda \left( {aA}\right) > \lambda \left( {aB}\right) \), and let \( A \in {V}_{2} \) prefer a neighbour \( a \) to a neighbour \( b \) iff \( \lambda \left( {aA}\right) < \lambda \left( {bA}\right) \) . Note that the total function defined by this assignment of preferences is at most \( k - 1 \) on every edge, since if \( \lambda \left( {aA}\right) = j \) then \( a \) prefers at most \( k - j \) of its neighbours to \( A \), and \( A \) prefers at most \( j - 1 \) of its neighbours to \( a \) . Hence, by Theorem \( {11}, G \) is \( k \) -choosable. | Yes |
Theorem 11 (Dual to Theorem 8). In order that a G-module A be cohomo-logically trivial, it is necessary and sufficient that there be an exact sequence \( 0 \rightarrow \mathrm{A} \rightarrow {\mathrm{I}}_{0} \rightarrow {\mathrm{I}}_{1} \rightarrow 0 \), where the \( {\mathrm{I}}_{i} \) are injective \( \mathbf{Z}\left\lbrack \mathrm{G}\right\rbrack \) -modules. | As before, there is an exact sequence\n\[
0 \rightarrow \mathrm{A} \rightarrow {\mathrm{I}}_{0} \rightarrow \mathrm{R} \rightarrow 0
\]\nwith \( {\mathrm{I}}_{0}\mathbf{Z}\left\lbrack \mathrm{G}\right\rbrack \) -injective. Since \( \mathrm{A} \) is cohomologically trivial, so is \( \mathrm{R} \) ; on the other hand, \( {\mathrm{I}}_{0} \) is \( \mathbf{Z} \) -injective (by lemma 7), hence \( \mathrm{R} \) is too. Theorem 10 then guarantees that \( \mathrm{R} \) is \( \mathbf{Z}\left\lbrack \mathrm{G}\right\rbrack \) -injective. | No |
Every non-orientable path connected CW n-manifold has an orientable path connected double cover. | Null | No |
Proposition 8.4.2. Ordinary reduction, supersingular reduction, and multiplicative reduction are well defined on equivalence classes of \( \mathfrak{p} \) -minimal Weierstrass equations. If \( E \) and \( {E}^{\prime } \) are equivalent \( \mathfrak{p} \) -minimal Weierstrass equations with good reduction at \( \mathfrak{p} \) then their reductions define isomorphic elliptic curves over \( {\overline{\mathbb{F}}}_{p} \) . | Proof. If \( E \) and \( {E}^{\prime } \) are equivalent then by Exercise 8.1.1(b)
\[
{u}^{12}{\Delta }^{\prime } = \Delta ,\;{u}^{4}{c}_{4}^{\prime } = {c}_{4},
\]
where \( u \) comes from the admissible change of variable taking \( E \) to \( {E}^{\prime } \) . Recall the disjoint union mentioned early in this section,
\[
{\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } = {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) }^{ * } \cup \mathfrak{p}{\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) }
\]
Assume \( {\Delta }^{\prime } \in {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) }^{ * } \) . If also \( \Delta \in \mathfrak{p}{\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \) then \( {u}^{12} \in \mathfrak{p}{\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \) and thus \( {u}^{4} \in \mathfrak{p}{\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \) so that \( {c}_{4} \in \mathfrak{p}{\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \) . This means that \( E \) has additive reduction, impossible since \( E \) is \( \mathfrak{p} \) -minimal. Thus there is no equivalence between equations of good and multiplicative reduction.
Given a change of variable between two equations of good reduction, we may further assume by Lemma 8.4.1 that \( u \in {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \), and so the relation \( {u}^{12}{\Delta }^{\prime } = \) \( \Delta \) shows that \( u \in {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) }^{ * } \) . Therefore \( \widetilde{u} \neq 0 \) in \( {\overline{\mathbb{F}}}_{p} \) . Also the other coefficients \( r, s, t \) from the admissible change of variable taking \( E \) to \( {E}^{\prime } \) lie in \( {\overline{\mathbb{Z}}}_{\left( \mathfrak{p}\right) } \), similarly to Exercise 8.3.3(b, c) (Exercise 8.4.3), so they reduce to \( {\overline{\mathbb{F}}}_{p} \) under (8.24). The two reduced Weierstrass equations differ by the reduced change of variable, giving the last statement of the proposition. Since isomorphic elliptic curves have the same \( p \) -torsion structure, this shows that ordinary and supersingular reduction are preserved under equivalence. | No |
Let \( D < 0 \) be a fundamental discriminant, and denote by \( h\left( D\right) \) the class number of \( K = \mathbb{Q}\left( \sqrt{D}\right) \) . Denote by \( \mathcal{Q}\left( D\right) \) the set of equivalence classes of quadratic numbers \( \tau = \left( {-b + \sqrt{D}}\right) /\left( {2a}\right) \) of discriminant \( D \), modulo the natural action of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), which has cardinality \( h\left( D\right) \) . We have the following formulas: | L\left( {{\chi }_{D},1}\right) = \frac{{2\pi h}\left( D\right) }{w\left( D\right) {\left| D\right| }^{1/2}},\;L\left( {{\chi }_{D},0}\right) = \frac{{2h}\left( D\right) }{w\left( D\right) },\]
\[
\frac{{L}^{\prime }\left( {{\chi }_{D},1}\right) }{L\left( {{\chi }_{D},1}\right) } = \gamma - \log \left( 2\right) - \frac{\log \left( \left| D\right| \right) }{2} - \frac{2}{h\left( D\right) }\mathop{\sum }\limits_{{\tau \in \mathcal{Q}\left( D\right) }}\log \left( {\Im {\left( \tau \right) }^{1/2}{\left| \eta \left( \tau \right) \right| }^{2}}\right) ,
\]
\[
\frac{{L}^{\prime }\left( {{\chi }_{D},0}\right) }{L\left( {{\chi }_{D},0}\right) } = \log \left( {4\pi }\right) - \frac{\log \left( \left| D\right| \right) }{2} + \frac{2}{h\left( D\right) }\mathop{\sum }\limits_{{\tau \in \mathcal{Q}\left( D\right) }}\log \left( {\Im {\left( \tau \right) }^{1/2}{\left| \eta \left( \tau \right) \right| }^{2}}\right) .
\] | Yes |
Theorem 6.3.1 Let \( X, Y \), and \( Z \) be graphs. If \( f : Z \rightarrow X \) and \( g : Z \rightarrow Y \), then there is a unique homomorphism \( \phi \) from \( Z \) to \( X \times Y \) such that \( f = {p}_{X} \circ \phi \) and \( g = {p}_{Y} \circ \phi \) . | Proof. Assume that we are given homomorphisms \( f : Z \rightarrow X \) and \( g : Z \rightarrow Y \) . The map\\ \[\\ \phi : z \mapsto \left( {f\left( z\right), g\left( z\right) }\right)\\ \] is readily seen to be a homomorphism from \( Z \) to \( X \times Y \) . Clearly, \( {p}_{X} \circ \phi = f \) and \( {p}_{Y} \circ \phi = g \), and furthermore, \( \phi \) is uniquely determined by \( f \) and \( g \) . \( ▱ \) | Yes |
If \( A \subseteq \mathbb{N} \) has positive upper density, then \( A \) contains infinitely many arithmetic progressions of length 3. | Null | No |
Corollary 1.141. A multimap \( F : X \rightrightarrows Y \) between two metric spaces is c-subregular at \( \left( {\bar{x},\bar{y}}\right) \in F \) if and only if \( {F}^{-1} \) is \( c \) -calm at \( \left( {\bar{y},\bar{x}}\right) \) . | Null | No |
We have \({B}_{n} = \mathop{\sum }\limits_{{\pi \in \bar{S}\left( n\right) }}W\left( \pi \right)\) | The proof of recurrence (6) for \( \mathop{\sum }\limits_{{\pi \in \bar{S}\left( n\right) }}W\left( \pi \right) \) is analogous, by considering separately the cases \( {\pi }_{n + 1} = n + 1 \) and \( {\pi }_{n + 1} \neq n + 1 \), and is left to the exercises. | No |
We have \[
\mathop{\sum }\limits_{\lambda }{s}_{\lambda }\left( \mathbf{x}\right) {s}_{\lambda }\left( \mathbf{y}\right) = \mathop{\prod }\limits_{{i, j \geq 1}}\frac{1}{1 - {x}_{i}{y}_{j}}.
\] | Note that just as the Robinson-Schensted correspondence is gotten by restricting Knuth's generalization to the case where all entries are distinct, we can obtain
\[
n! = \mathop{\sum }\limits_{{\lambda \vdash n}}{\left( {f}^{\lambda }\right) }^{2}
\]
by taking the coefficient of \( {x}_{1}\cdots {x}_{n}{y}_{1}\cdots {y}_{n} \) on both sides of Theorem 4.8.4. | No |
Proposition 3.4. Let \( B \) be a \( \Lambda \) -module and \( \left\{ {A}_{j}\right\}, j \in J \) be a family of \( \Lambda \) - modules. Then there is an isomorphism\n\[
\eta : {\operatorname{Hom}}_{A}\left( {{\bigoplus }_{j \in J}{A}_{j}, B}\right) \rightarrow \mathop{\prod }\limits_{{j \in J}}{\operatorname{Hom}}_{A}\left( {{A}_{j}, B}\right) .
\] | The proof reveals that this theorem is merely a restatement of the universal property of the direct sum. For \( \psi : {\bigoplus }_{j \in J}{A}_{j} \rightarrow B \), define \( \eta \left( \psi \right) = {\left( \psi {\iota }_{j} : {A}_{j} \rightarrow B\right) }_{j \in J} \) . Conversely a family \( \left\{ {{\psi }_{j} : {A}_{j} \rightarrow B}\right\}, j \in J \), gives rise to a unique map \( \psi : {\bigoplus }_{j \in J}{A}_{j} \rightarrow B \) . The projections \( {\pi }_{j} : \mathop{\prod }\limits_{{j \in J}}{\operatorname{Hom}}_{A}\left( {{A}_{j}, B}\right) \) \( \rightarrow {\operatorname{Hom}}_{\Lambda }\left( {{A}_{j}, B}\right) \) are given by \( {\pi }_{j}\eta = {\operatorname{Hom}}_{\Lambda }\left( {{\iota }_{j}, B}\right) \) . | Yes |
Proposition 15.36. Let the notation be as in Proposition 13.32. Then \( {\varepsilon }_{j}{\mathcal{X}}_{\infty } \) has no nonzero finite submodules. | Null | No |
Theorem 10.5. Let \( D \) be a domain in \( \mathbb{C} \) and suppose that \( \left\{ {f}_{n}\right\} \) is a sequence in \( \mathbf{H}\left( D\right) \) with \( {f}_{n} \) not identically 0 for all \( n \) . | For part (a), we only have to verify the formula for the order of \( z \) . We note that the sum in that formula is finite (i.e., all but finitely many summands are zero). Let \( {z}_{0} \in D \) and let \( K \subset D \) be a compact set containing a neighborhood of \( {z}_{0} \) . There is an \( N \) in \( {\mathbb{Z}}_{ > 0} \) such that \( \left| {1 - {f}_{n}\left( z\right) }\right| < \frac{1}{2} \) for all \( z \in K \) and all \( n \geq N \) . Therefore, \( {f}_{n}\left( z\right) \neq 0 \) for all \( z \in K \) and for all \( n \geq \widetilde{N} \) . Thus \[
{v}_{{z}_{0}}\left( f\right) = {v}_{{z}_{0}}\left( {\mathop{\prod }\limits_{{n = 1}}^{{N - 1}}{f}_{n}}\right) + {v}_{{z}_{0}}\left( {\mathop{\prod }\limits_{{n = N}}^{\infty }{f}_{n}}\right) = \mathop{\sum }\limits_{{n = 1}}^{{N - 1}}{v}_{{z}_{0}}\left( {f}_{n}\right) + 0.
\] | No |
Proposition 12.6. For a group extension \( G\overset{\kappa }{ \rightarrow }E\overset{\rho }{ \rightarrow }Q \) the following conditions are equivalent: | Proof. (1) implies (2). If (1) holds, then \( {p}_{a} = \mu \left( a\right) \) is a cross-section of \( E \) , relative to which \( {s}_{a, b} = 1 \) for all \( a, b \), since \( \mu \left( a\right) \mu \left( b\right) = \mu \left( {ab}\right) \) .
(2) implies (3). If \( {s}_{a, b} = 1 \) for all \( a, b \), then \( \varphi : Q \rightarrow \operatorname{Aut}\left( G\right) \) is a homomorphism, by \( \left( A\right) \), and \( \left( M\right) \) shows that \( E\left( {s,\varphi }\right) = G{ \rtimes }_{\varphi }Q \) . Then \( E \) is equivalent to \( E\left( {s,\varphi }\right) \), by Schreier’s theorem.
(3) implies (4). A semidirect product \( G{ \rtimes }_{\psi }Q \) of \( G \) by \( Q \) is a group extension \( E\left( {t,\psi }\right) \) in which \( {t}_{a, b} = 1 \) for all \( a, b \) . If \( E \) is equivalent to \( G{ \rtimes }_{\psi }Q \), then, relative to any cross-section of \( E, E\left( {s,\varphi }\right) \) and \( E\left( {t,\psi }\right) \) are equivalent, and \( \left( E\right) \) yields \( {s}_{a, b} = {u}_{a}{}_{\psi }^{a}{u}_{b}{t}_{a, b}{u}_{ab}^{-1} = {}_{\varphi }^{a}{u}_{b}{u}_{a}{u}_{ab}^{-1} \) for all \( a, b \in Q \) .
(4) implies (1). If \( {s}_{a, b} = {}^{a}{u}_{b}{u}_{a}{u}_{ab}^{-1} \) for all \( a, b \in Q \), then \( {u}_{a}^{-1}{}^{a}\left( {u}_{b}^{-1}\right) {s}_{a, b} \) \( = {u}_{ab}^{-1} \) and \( \mu : a \mapsto \kappa \left( {u}_{a}^{-1}\right) {p}_{a} \) is a homomorphism, since
\[
\mu \left( a\right) \mu \left( b\right) = \kappa \left( {{u}_{a}^{-1}{}^{a}\left( {u}_{b}^{-1}\right) {s}_{a, b}}\right) {p}_{ab} = \kappa \left( {u}_{ab}^{-1}\right) {p}_{ab} = \mu \left( {ab}\right) .▱
\] | Yes |
Let \( E \) and \( F \) be affine spaces, and \( A : E \rightarrow F \) a multivalued map whose graph \( \operatorname{gr}\left( A\right) = C \) is a nonempty convex set in \( E \times F \) . Then | The inclusion follows immediately from Lemma 5.13. If \( x \in \operatorname{rai}\operatorname{dom}\left( A\right) \) and \( y \in \operatorname{rai}A\left( x\right) \neq \varnothing \), then Lemma 5.13 implies that \( \left( {x, y}\right) \in \operatorname{raigr}\left( A\right) \), and we have \( x \in \operatorname{dom}\operatorname{rai}\left( {\operatorname{gr}\left( A\right) }\right) \) . If \( x \in \operatorname{rai}\operatorname{dom}\left( A\right) \) and \( F \) is finite-dimensional, then Lemma 5.3 implies that \( \operatorname{rai}A\left( x\right) \neq \varnothing \) . | Yes |
Let \( p \) be a prime number and \( \alpha \) an algebraic number. The following conditions are equivalent: | Clear and left to the reader (Exercise 7). | No |
Suppose \( X \) is a locally convex space over \( \mathbb{R} \) or \( \mathbb{C} \). Then the topology of \( X \) is given by a directed family of seminorms. This family can be chosen to be countable if \( X \) is first countable. | Proof. \( {\mathcal{B}}_{1} \) exists by Proposition 3.1. By the way, Reed and Simon [29] define a locally convex space this way. What does one do if the family is not directed? There is a standard construction that goes as follows, if \( {\mathcal{F}}_{0} \) is any family of seminorms. 1. If \( {\mathcal{F}}_{0} \) is finite, set \( \mathcal{F} = \left\{ {\mathop{\sum }\limits_{{p \in {\mathcal{F}}_{0}}}p}\right\} \). 2. If \( {\mathcal{F}}_{0} \) is countably infinite, write \( {\mathcal{F}}_{0} = \left\{ {{p}_{1},{p}_{2},{p}_{3},\ldots }\right\} \), and set \[ \mathcal{F} = \left\{ {\mathop{\sum }\limits_{{j = 1}}^{n}{p}_{j} : n = 1,2,\ldots }\right\} \] \[ = \left\{ {{p}_{1},{p}_{1} + {p}_{2},{p}_{1} + {p}_{2} + {p}_{3},\ldots }\right\} \text{.} \] 3. If \( {\mathcal{F}}_{0} \) is uncountable, set \[ \mathcal{F} = \left\{ {\mathop{\sum }\limits_{{p \in F}}p : F\text{ is a finite subset of }{\mathcal{F}}_{0}}\right\} . \] Suppose \( {x}_{\alpha } \rightarrow x \) in the \( \mathcal{F} \) -topology, where \( \left\langle {x}_{\alpha }\right\rangle \) is a net, and \( \mathcal{F} \) is defined above. Then \( p\left( {{x}_{\alpha } - x}\right) \rightarrow 0 \) in \( \mathbb{R} \) for all \( p \in \mathcal{F} \) since each \( p \in \mathcal{F} \) is continuous. Hence \( p\left( {{x}_{\alpha } - x}\right) \rightarrow 0 \) for all \( p \in {\mathcal{F}}_{0} \) by squeezing. On the other hand, if \( \left\langle {x}_{\alpha }\right\rangle \) is a net in \( X \), and \( x \in X \), and \( p\left( {{x}_{\alpha } - x}\right) \rightarrow 0 \) for all \( p \in {\mathcal{F}}_{0} \), then \( p\left( {{x}_{\alpha } - x}\right) \rightarrow 0 \) for all \( p \in \mathcal{F} \) (finite sums), so that for all \( n \in \mathbb{N} \), there exists \( \beta \) such that \( \alpha \succ \beta \Rightarrow p\left( {{x}_{\alpha } - x}\right) < {2}^{-n} \), that is \( {x}_{\alpha } \in x + B\left( {p,{2}^{-n}}\right) \). That is, \( {x}_{\alpha } \rightarrow x \) in the topology induced by \( \mathcal{F} \) if and only if \( p\left( {{x}_{\alpha } - x}\right) \rightarrow 0 \) for all \( p \in {\mathcal{F}}_{0} \). In particular, the convergent nets [and thus the topology, by Proposition 1.3(a)] does not depend on the ordering of the seminorms in Case 2 above. | Yes |
Show that \( \mathbb{Z}\left\lbrack \rho \right\rbrack /\left( \lambda \right) \) has order 3. | Null | No |
Proposition 4.3.11 Let \( A \subseteq \left\lbrack {0,1}\right\rbrack \) be a strong measure zero set and \( f \) : \( \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \) a continuous map. Then the set \( f\left( A\right) \) has strong measure zero. | Proof. Let \( \left( {a}_{n}\right) \) be any sequence of positive real numbers. We have to show that there exist open intervals \( {J}_{n}, n \in \mathbb{N} \), such that \( \left| {J}_{n}\right| \leq {a}_{n} \) and \( f\left( A\right) \subseteq \mathop{\bigcup }\limits_{n}{J}_{n} \) . Since \( f \) is uniformly continuous, for each \( n \) there is a positive real number \( {b}_{n} \) such that whenever \( X \subseteq \left\lbrack {0,1}\right\rbrack \) is of diameter at most \( {b}_{n} \) , the diameter of \( f\left( X\right) \) is at most \( {a}_{n} \) . Since \( A \) has strong measure zero, there are open intervals \( {I}_{n}, n \in \mathbb{N} \), such that \( \left| {I}_{n}\right| \leq {b}_{n} \) and \( A \subseteq \mathop{\bigcup }\limits_{n}{I}_{n} \) . Take \( {J}_{n} = f\left( {I}_{n}\right) \) . | Yes |
Let \( G \) be a compact Abelian group and let \( {L}_{a} \) be the Koopman operator induced by the rotation by \( a \in G \) . Since every character \( \chi \in {G}^{ * } \) is an eigenfunction of \( {L}_{a} \) corresponding to the eigenvalue \( \chi \left( a\right) \in \mathbb{T} \) and since \( \operatorname{lin}{G}^{ * } \) is dense in \( \mathrm{C}\left( G\right) \) (Proposition 14.7), \( {L}_{a} \) has discrete spectrum on \( \mathrm{C}\left( G\right) \) . A fortiori, \( {L}_{a} \) has discrete spectrum also on \( {\mathrm{L}}^{p}\left( G\right) \) for every \( 1 \leq p < \infty \) . | Direct proof of Theorem 17.11. Let \( T \) be the Koopman operator of the ergodic system \( \left( {\mathrm{X};\varphi }\right) \) with discrete spectrum, and let \( \Gamma \mathrel{\text{:=}} {\sigma }_{\mathrm{p}}\left( T\right) \) be its point spectrum. By Proposition 7.18, each eigenvalue is unimodular and simple, and \( \Gamma \) is a subgroup of \( \mathbb{T} \) . Each eigenfunction is unimodular up to a multiplicative constant. | No |
Theorem 5. Let \( X = X\left( \omega \right) \) be a random element with values in the Borel space \( \left( {E,\mathcal{E}}\right) \) . Then there is a regular conditional distribution of \( X \) with respect to \( \mathcal{G} \subseteq \mathcal{F} \) . | Let \( \varphi = \varphi \left( e\right) \) be the function in Definition 9. By (2) in this definition \( \varphi \left( {X\left( \omega \right) }\right) \) is a random variable. Hence, by Theorem 4, we can define the conditional distribution \( Q\left( {\omega ;A}\right) \) of \( \varphi \left( {X\left( \omega \right) }\right) \) with respect to \( \mathcal{G}, A \in \varphi \left( E\right) \cap \mathcal{B}\left( R\right) \) .
We introduce the function \( \widetilde{Q}\left( {\omega ;B}\right) = Q\left( {\omega ;\varphi \left( B\right) }\right), B \in \mathcal{E} \) . By (3) of Definition 9, \( \varphi \left( B\right) \in \varphi \left( E\right) \cap \mathcal{B}\left( R\right) \) and consequently \( \widetilde{Q}\left( {\omega ;B}\right) \) is defined. Evidently \( \widetilde{Q}\left( {\omega ;B}\right) \) is a measure in \( B \in \mathcal{E} \) for every \( \omega \) . Now fix \( B \in \mathcal{E} \) . By the one-to-one character of the mapping \( \varphi = \varphi \left( e\right) \),
\[
\widetilde{Q}\left( {\omega ;B}\right) = Q\left( {\omega ;\varphi \left( B\right) }\right) = \mathsf{P}\{ \varphi \left( X\right) \in \varphi \left( B\right) \mid \mathcal{G}\} \left( \omega \right) = \mathsf{P}\{ X \in B \mid \mathcal{G}\} \left( \omega \right) \;\text{ (a. s.). }
\]
Therefore \( \widetilde{Q}\left( {\omega ;B}\right) \) is a regular conditional distribution of \( X \) with respect to \( \mathcal{G} \) | Yes |
Let \( \left( {U, x}\right) \) be a chart around \( p \) . Then any tangent vector \( v \in {M}_{p} \) can be uniquely written as a linear combination \( v = \mathop{\sum }\limits_{i}{\alpha }_{i}\partial /\partial {x}^{i}\left( p\right) \) . In fact, \( {\alpha }_{i} = v\left( {x}^{i}\right) \) . | We may assume without loss of generality that \( x\left( p\right) = 0 \), and that \( x\left( U\right) \) is star-shaped. By Lemma 3.1, any \( f \in \mathcal{F}M \) satisfies \( f \circ {x}^{-1} = f\left( p\right) + \sum {u}^{i}{\psi }_{i} \), with \( {\psi }_{i}\left( 0\right) = \partial /\partial {x}^{i}\left( p\right) \left( f\right) \) . Thus, \( {\left. f\right| }_{U} = f\left( p\right) + \mathop{\sum }\limits_{i}{x}^{i}\left( {{\psi }_{i} \circ x}\right) {\left. \right| }_{U} \) , and
\[
v\left( f\right) = v\left( {f\left( p\right) }\right) + \mathop{\sum }\limits_{i}\left\lbrack {v\left( {x}^{i}\right) \cdot {\psi }_{i}\left( 0\right) + {x}^{i}\left( p\right) \cdot v\left( {{\psi }_{i} \circ x}\right) }\right\rbrack = \mathop{\sum }\limits_{i}v\left( {x}^{i}\right) \frac{\partial }{\partial {x}^{i}}\left( p\right) \left( f\right) ,
\]
where we have used the result of Exercise 5 below. It remains to show that the \( \partial /\partial {x}^{i}\left( p\right) \) are linearly independent; observe that
\[
\frac{\partial }{\partial {x}^{i}}\left( p\right) \left( {x}^{j}\right) = {D}_{i}\left( {{x}^{j} \circ {x}^{-1}}\right) \left( 0\right) = {D}_{i}\left( {u}^{j}\right) \left( 0\right) = {\delta }_{ij}.
\]
Thus, if \( \sum {\alpha }_{i}\partial /\partial {x}^{i}\left( p\right) = 0 \), then \( 0 = \sum {\alpha }_{i}\partial /\partial {x}^{i}\left( p\right) \left( {x}^{j}\right) = {\alpha }_{j} \) . | Yes |
Theorem 1.29. Every bounded linear functional \( \Lambda \) on a Hilbert space \( \mathcal{H} \) is given by inner product with a (unique) fixed vector \( {h}_{0} \) in \( \mathcal{H} : \Lambda \left( h\right) = \left\langle {h,{h}_{0}}\right\rangle \) . Moreover, the norm of the linear functional \( \Lambda \) is \( \begin{Vmatrix}{h}_{0}\end{Vmatrix} \). | Suppose \( \Lambda \) is a bounded linear functional on \( \mathcal{H} \) . If \( \Lambda \) is identically 0, choose \( {h}_{0} = 0 \) . Otherwise, set
\[
M = \ker \Lambda \equiv \{ h \in \mathcal{H} : \Lambda \left( h\right) = 0\} .
\]
Since \( \Lambda \) is linear, \( M \) is a subspace of \( \mathcal{H} \), and since \( \Lambda \) is continuous, \( M = {\Lambda }^{-1}\left( 0\right) \) is closed. Note that \( M \neq \mathcal{H} \) since we are assuming \( \Lambda \neq 0 \) . Pick a nonzero vector \( z \in {M}^{ \bot } \) . By scaling if necessary we may assume \( \Lambda \left( z\right) = 1 \) . Consider, for arbitrary \( h \in \mathcal{H} \), the vector \( \Lambda \left( h\right) z - h \) and observe that if we apply \( \Lambda \) to this vector we get 0, i.e., it lies in \( M \) . Since \( z \) was chosen to lie in \( {M}^{ \bot } \), this says
\[
\Lambda \left( h\right) z - h \bot z
\]
so that for every \( h \in \mathcal{H} \) ,
\[
\langle \Lambda \left( h\right) z - h, z\rangle = 0.
\]
Rearranging this last line we see that \( \Lambda \left( h\right) = \langle h, z/\parallel z{\parallel }^{2}\rangle \), which gives the existence statement with \( {h}_{0} = z/\parallel z{\parallel }^{2} \) . Uniqueness is immediate, and since we have already observed that \( \parallel \mathbf{\Lambda }\parallel = \begin{Vmatrix}{h}_{0}\end{Vmatrix} \), we are done. | Yes |
Corollary 1.4. Theorem 1.1 follows from Theorem 1.3. | Proof. By the mean value theorem there exists \( \bar{x} \in \left( {a,{s}_{n - 1}}\right) \) such that\n\[
{\int }_{a}^{{s}_{n - 1}}{f}^{\left( n\right) }\left( {s}_{n}\right) d{s}_{n} = {f}^{\left( n\right) }\left( \bar{x}\right) \left( {{s}_{n - 1} - a}\right) = {\int }_{a}^{{s}_{n - 1}}{f}^{\left( n\right) }\left( \bar{x}\right) d{s}_{n}.
\]\nThe proof is completed by substituting this in the iterated integral in the statement of Theorem 1.3 and using (1.3). | No |
Let \( V \) be a one-dimensional analytic subvariety of an open subset of \( {\mathbb{C}}^{n} \) . Then \( \operatorname{area}\left( V\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\text{ area-with-multiplicity }\left( {{z}_{j}\left( V\right) }\right) \). | This is a local result and so we can assume that \( V \) can be parameterized by a one-one analytic map \( f : W \rightarrow V \subseteq {\mathbb{C}}^{n} \), where \( W \) is a domain in the complex plane. Let \( \zeta \in W \) be \( \zeta = s + {it} \) and let \( f = \left( {{f}_{1},{f}_{2},\ldots ,{f}_{n}}\right) \), where \( {f}_{k} = \) \( {u}_{k} + i{v}_{k} \) . Set \( X\left( {s, t}\right) = f\left( \zeta \right) \) and view \( X \) as a map of \( W \) into \( {\mathbb{R}}^{2n} \) . The classical formula for the area of the map \( X \) is \( {\int }_{W}\left| {X}_{s}\right| \left| {X}_{t}\right| \sin \left( \theta \right) {dsdt} \), where \( \theta \) is the angle between \( {X}_{s} \) and \( {X}_{t} \) . We have \( {X}_{s} = \left( {{f}_{1}^{\prime },{f}_{2}^{\prime },\ldots ,{f}_{n}^{\prime }}\right) \) and \( {X}_{t} = \left( {i{f}_{1}^{\prime }, i{f}_{2}^{\prime },\ldots, i{f}_{n}^{\prime }}\right) \) by the Cauchy-Riemann equations. Hence \( \left| {X}_{t}\right| = \left| {X}_{s}\right| \) and the vectors \( {X}_{s} \) and \( {X}_{t} \) are orthogonal in \( {\mathbb{R}}^{2n} \), and so \( \sin \left( \theta \right) \equiv 1 \) . Thus the previous formula for the area of the image of \( X \) becomes \( \operatorname{area}\left( V\right) = {\int }_{W}\left( {\mathop{\sum }\limits_{j}{\left| {f}_{j}^{\prime }\right| }^{2}}\right) {dsdt} = \mathop{\sum }\limits_{j}{\int }_{W}{\left| {f}_{j}^{\prime }\right| }^{2}{dsdt} \) . Since \( {\int }_{W}{\left| {f}_{j}^{\prime }\right| }^{2}{dsdt} \) is the area-with-multiplicity of \( {f}_{j}\left( W\right) \), this completes the proof. | Yes |
Corollary 1. Every irreducible representation \( {\mathrm{W}}_{i} \) is contained in the regular representation with multiplicity equal to its degree \( {n}_{i} \). | According to th. 4, this number is equal to \( \left\langle {{r}_{\mathrm{G}},{\chi }_{i}}\right\rangle \), and we have\\[2mm] \left\langle {{r}_{\mathrm{G}},{\chi }_{i}}\right\rangle = \frac{1}{g}\mathop{\sum }\limits_{{s \in \mathrm{G}}}{r}_{\mathrm{G}}\left( {s}^{-1}\right) {\chi }_{i}\left( s\right) = \frac{1}{g}g \cdot {\chi }_{i}\left( 1\right) = {\chi }_{i}\left( 1\right) = {n}_{i}.\\[2mm] | Yes |
Theorem 1.8 Let \( T \) be a compact operator from \( E \) to \( E \) . | 1. Suppose that 0 is not a spectral value of \( T \) . Then \( I = T{T}^{-1} \) is a compact operator by Proposition 1.2. By the Riesz Theorem (page 49), this implies that \( E \) is finite-dimensional.
2. Take \( \lambda \in {\mathbb{K}}^{ * } \) . Then \( \lambda \) is an eigenvalue of \( T \) if and only if \( I - T/\lambda \) is not injective, and \( \ker \left( {{\lambda I} - T}\right) = \ker \left( {I - T/\lambda }\right) \) . On the other hand, \( \lambda \) is a spectral value of \( T \) if and only if \( I - T/\lambda \) is not invertible in \( L\left( E\right) \) . Thus it suffices to apply Proposition 1.6 to prove assertion 2.
3. For assertion 3, it is enough to show that, for every \( \varepsilon > 0 \), there is only a finite number (perhaps 0) of spectral values \( \lambda \) of \( T \) such that \( \left| \lambda \right| \geq \varepsilon \) . Suppose, on the contrary, that, for a certain \( \varepsilon > 0 \), there exists a sequence \( {\left( {\lambda }_{n}\right) }_{n \in \mathbb{N}} \) of pairwise distinct spectral values of \( T \) such that \( \left| {\lambda }_{n}\right| \geq \varepsilon \) for every \( n \in \mathbb{N} \) . By part 2, all the \( {\lambda }_{n} \) are eigenvalues of \( T \) . Thus there exists a sequence \( \left( {e}_{n}\right) \) of elements of \( E \) of norm 1 such that \( T{e}_{n} = {\lambda }_{n}{e}_{n} \) for every \( n \in \mathbb{N} \) . Since the eigenvalues \( {\lambda }_{n} \) are pairwise distinct, it is easy to see (and it is a classical result) that the family \( {\left\{ {e}_{n}\right\} }_{n \in \mathbb{N}} \) is linearly independent. For each \( n \in \mathbb{N} \), let \( {E}_{n} \) be the span of the \( n + 1 \) first vectors \( {e}_{0},\ldots ,{e}_{n} \) . The sequence \( {\left( {E}_{n}\right) }_{n \in \mathbb{N}} \) is then a strictly increasing sequence of finite-dimensional spaces. By Lemma 1.7, there exists a sequence \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \) of vectors of norm 1 such that, for every integer \( n \in \mathbb{N} \),
\[
{u}_{n} \in {E}_{n + 1}\;\text{ and }\;d\left( {{u}_{n},{E}_{n}}\right) \geq \frac{1}{2}
\]
(in fact, since \( {E}_{n} \) has finite dimension, we could replace \( \frac{1}{2} \) by 1 here). Define \( {v}_{n} = {\lambda }_{n + 1}^{-1}{u}_{n} \) . The sequence \( \left( {v}_{n}\right) \) is bounded by \( 1/\varepsilon \) . Moreover, if \( n > m \),
\[
T{v}_{n} - T{v}_{m} = {u}_{n} - {v}_{n, m}\;\text{ with }\;{v}_{n, m} = T{v}_{m} + \frac{1}{{\lambda }_{n + 1}}\left( {{\lambda }_{n + 1}I - T}\right) {u}_{n}.
\]
But \( T{v}_{m} \in {E}_{m + 1} \subset {E}_{n} \) and \( \left( {{\lambda }_{n + 1}I - T}\right) \left( {E}_{n + 1}\right) \subset {E}_{n} \) . Thus \( {v}_{n, m} \in {E}_{n} \) and \( \begin{Vmatrix}{T{v}_{n} - T{v}_{m}}\end{Vmatrix} \geq \frac{1}{2} \), contradicting the compactness of \( T \) (the sequence \( {\left( {v}_{n}\right) }_{n \in \mathbb{N}} \) is bounded and its image under \( T \) has no Cauchy subsequence, hence no convergent subsequence). | Yes |
Proposition 9.19. If \( \Gamma \leq {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) is a lattice, the hyperbolic measure defined by the volume form \( \mathrm{d}m = \frac{1}{{y}^{2}}\mathrm{\;d}x\mathrm{\;d}y\mathrm{\;d}\theta \) in Lemma 9.16 induces a \( {fi} \) - nite \( {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) -invariant measure \( {m}_{X} \) on \( X = \Gamma \smallsetminus {\mathrm{{PSL}}}_{2}\left( \mathbb{R}\right) \) . | In fact if \[ \pi : {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \rightarrow X \] is the canonical quotient map \( \pi \left( g\right) = {\Gamma g} \) for \( g \in {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) and \( F \) is a finite volume fundamental domain, then \[ {m}_{X}\left( B\right) = m\left( {F \cap {\pi }^{-1}B}\right) \] for \( B \subseteq X \) measurable defines the \( {\operatorname{PSL}}_{2}\left( \mathbb{R}\right) \) -invariant measure on \( X \) . | Yes |
Corollary 14.23 Suppose that \( S \) is a lower semibounded self-adjoint extension of the densely defined lower semibounded symmetric operator \( T \) on \( \mathcal{H} \) . Let \( \lambda \in \mathbb{R} \) , \( \lambda < {m}_{T} \), and \( \lambda \leq {m}_{S} \) . Then \( S \) is equal to the Friedrichs extension \( {T}_{F} \) of \( T \) if and only if \( \mathcal{D}\left\lbrack S\right\rbrack \cap \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) = \{ 0\} \) . | Since then \( \lambda \in \rho \left( {T}_{F}\right) \), Example 14.6, with \( A = {T}_{F},\mu = \lambda \), yields a boundary triplet \( \left( {\mathcal{K},{\Gamma }_{0},{\Gamma }_{1}}\right) \) for \( {T}^{ * } \) such that \( {T}_{0} = {T}_{F} \) . By Propositions 14.7(v) and 14.21, there is a self-adjoint relation \( \mathcal{B} \) on \( \mathcal{K} \) such that \( S = {T}_{\mathcal{B}} \) and \( \mathcal{B} - M\left( \lambda \right) \geq 0 \) . Thus, all assumptions of Theorem 14.22 are fulfilled. Recall that \( \mathcal{K} = \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) . Therefore, by (14.56), \( \mathcal{D}\left\lbrack {T}_{\mathcal{B}}\right\rbrack \cap \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) = \{ 0\} \) if and only if \( \gamma \left( \lambda \right) \mathcal{D}\left\lbrack {\mathcal{B} - M\left( \lambda \right) }\right\rbrack = \{ 0\} \) , that is, \( \mathcal{D}\left\lbrack {\mathcal{B} - M\left( \lambda \right) }\right\rbrack = \{ 0\} \) . By (14.56) and (14.57) the latter is equivalent to \( {\mathfrak{t}}_{{T}_{\mathcal{B}}} = {\mathfrak{t}}_{{T}_{F}} \) and so to \( {T}_{\mathcal{B}} = {T}_{F} \) . (Note that we even have \( M\left( \lambda \right) = 0 \) and \( \gamma \left( \lambda \right) = {I}_{\mathcal{H}} \upharpoonright \mathcal{K} \) by Example 14.12.) | Yes |
Using Gram-Schmidt orthogonalization, show multiplication induces a diffeomorphism of \( O\left( n\right) \times A \times N \) onto \( {GL}\left( {n,\mathbb{R}}\right) \) . | Theorem 1.15. Let \( G \) be a connected Lie group and \( U \) a neighborhood of \( e \) . Then \( U \) generates \( G \), i.e., \( G = { \cup }_{n = 1}^{\infty }{U}^{n} \) where \( {U}^{n} \) consists of all \( n \) -fold products of elements of \( U \ ). | No |
Find all integer solutions to the equation \( {x}^{2} + {11} = {y}^{3} \) . | In the ring \( \mathbb{Z}\left\lbrack {\left( {1 + \sqrt{-{11}}}\right) /2}\right\rbrack \), we can factor the equation as
\[
\left( {x - \sqrt{-{11}}}\right) \left( {x + \sqrt{-{11}}}\right) = {y}^{3}.
\]
Now, suppose that \( \delta \left| {\left( {x - \sqrt{-{11}}}\right) \text{and}\delta }\right| \left( {x + \sqrt{-{11}}}\right) \) (which implies that \( \delta \mid y) \) . Then \( \delta \left| {{2x}\text{and}\delta }\right| 2\sqrt{-{11}} \) which means that \( \delta \mid 2 \) because otherwise, \( \delta \mid \sqrt{-{11}} \), meaning that \( {11} \mid x \) and \( {11} \mid y \), which we can see is not true by considering congruences \( {\;\operatorname{mod}\;{11}^{2}} \) . Then \( \delta = 1 \) or 2, since 2 has no factorization in this ring. We will consider these cases separately.
Case 1. \( \delta = 1 \) .
Then the two factors of \( {y}^{3} \) are coprime and we can write
\[
\left( {x + \sqrt{-{11}}}\right) = \varepsilon {\left( \frac{a + b\sqrt{-{11}}}{2}\right) }^{3},
\]
where \( a, b \in \mathbb{Z} \) and \( a \equiv b\left( {\;\operatorname{mod}\;2}\right) \) . Since the units of \( \mathbb{Z}\left\lbrack {\left( {1 + \sqrt{-{11}}}\right) /2}\right\rbrack \) are \( \pm 1 \), which are cubes, then we can bring the unit inside the brackets and rewrite the above without \( \varepsilon \) . We have
\[
8\left( {x + \sqrt{-{11}}}\right) = {\left( a + b\sqrt{-{11}}\right) }^{3} = {a}^{3} + {3a}{b}^{2}\sqrt{-{11}} - {33a}{b}^{2} - {11}{b}^{3}\sqrt{-{11}}
\]
and so, comparing real and imaginary parts, we get
\[
{8x} = {a}^{3} - {33a}{b}^{2} = a\left( {{a}^{2} - {33}{b}^{2}}\right) ,
\]
\[
8 = 3{a}^{2}b - {11}{b}^{3} = b\left( {3{a}^{2} - {11}{b}^{2}}\right) .
\]
This implies that \( b \mid 8 \) and so we have 8 possibilities: \( b = \pm 1, \pm 2, \pm 4, \pm 8 \) . Substituting these back into the equations to find \( a, x \), and \( y \), and remembering that \( a \equiv b\left( {\;\operatorname{mod}\;2}\right) \) and that \( a, x, y \in \mathbb{Z} \) will give all solutions to the equation.
Case 2. \( \delta = 2 \) .
If \( \delta = 2 \), then \( y \) is even and \( x \) is odd. We can write \( y = 2{y}_{1} \), which gives the equation
\[
\left( \frac{x + \sqrt{-{11}}}{2}\right) \left( \frac{x - \sqrt{-{11}}}{2}\right) = 2{y}_{1}^{3}.
\]
Since 2 divides the right-hand side of this equation, it must divide the left-hand side, so
\[
2\left| {\;\left( \frac{x + \sqrt{-{11}}}{2}\right) }\right.
\]
or
\[
2\left| {\;\left( \frac{x - \sqrt{-{11}}}{2}\right) .}\right.
\]
However, since \( x \) is odd,2 divides neither of the factors above. We conclude that \( \delta \neq 2 \), and thus we found all the solutions to the equation in our discussion of Case 1. | No |
The set \( D = \left\{ {\alpha < \kappa : {\bar{d}}_{\beta } \in {B}_{\alpha }}\right. \) for all \( \left. {\beta < \alpha }\right\} \) is closed unbounded. | It is easy to see that \( D \) is closed. Let \( {\alpha }_{0} < \kappa \) . Build a sequence \( {\alpha }_{0} < {\alpha }_{1} < \ldots \) such that for all \( \beta < {\alpha }_{n},{\bar{d}}_{\beta } \in {B}_{{\alpha }_{n + 1}} \) . If \( \alpha = \sup {\alpha }_{i} \), then \( \alpha \in D \) . | No |
Suppose \( \rho \) is a density matrix on \( \mathbf{H} \). Then the map \( {\Phi }_{\rho } : \mathcal{B}\left( \mathbf{H}\right) \rightarrow \mathbb{C} \) given by \( {\Phi }_{\rho }\left( A\right) = \operatorname{trace}\left( {\rho A}\right) = \operatorname{trace}\left( {A\rho }\right) \) is a family of expectation values. | If we define \( {\Phi }_{\rho }\left( A\right) = \operatorname{trace}\left( {\rho A}\right) \), then \( {\Phi }_{\rho }\left( I\right) = \operatorname{trace}\left( \rho \right) = 1 \). For any \( A \in \mathcal{B}\left( \mathbf{H}\right) \), we have, \[ \operatorname{trace}\left( {\rho {A}^{ * }}\right) = \operatorname{trace}\left( {{A}^{ * }\rho }\right) = \operatorname{trace}\left( {\left( \rho A\right) }^{ * }\right) = \overline{\operatorname{trace}\left( {\rho A}\right) }.\] It follows that \( \operatorname{trace}\left( {\rho A}\right) \) is real when \( A \) is self-adjoint. Let \( {\rho }^{1/2} \) be the nonnegative self-adjoint square root of \( \rho \). Then \( {\rho }^{1/2} \) and \( A{\rho }^{1/2} \) are Hilbert-Schmidt (in the latter case, by Point 3 of Proposition 19.3). It follows that \( \operatorname{trace}\left( {A{\rho }^{1/2}{\rho }^{1/2}}\right) = \operatorname{trace}\left( {{\rho }^{1/2}A{\rho }^{1/2}}\right) \), by Proposition 19.5. Thus, if \( A \) is self-adjoint and non-negative, \[ \operatorname{trace}\left( {\rho A}\right) = \operatorname{trace}\left( {{\rho }^{1/2}{\rho }^{1/2}A}\right) = \operatorname{trace}\left( {{\rho }^{1/2}A{\rho }^{1/2}}\right) \geq 0, \] (19.1) because \( {\rho }^{1/2}A{\rho }^{1/2} \) is self-adjoint and non-negative. We have established that \( {\Phi }_{\rho } \) satisfies Points 1,2, and 3 of Definition 19.6. Meanwhile, suppose \( {A}_{n}\psi \) converges in norm to \( {A\psi } \), for each \( \psi \) in \( \mathbf{H} \). Then \( \begin{Vmatrix}{{A}_{n}\psi }\end{Vmatrix} \) is bounded as a function of \( n \) for each fixed \( \psi \). Thus, by the principle of uniform boundedness (Theorem A.40), there is a constant \( C \) such that \( \begin{Vmatrix}{A}_{n}\end{Vmatrix} \leq C \). Now, if \( \left\{ {e}_{j}\right\} \) is an orthonormal basis for \( \mathbf{H} \), we have \[ \left| \left\langle {{e}_{j},{\rho }^{1/2}{A}_{n}{\rho }^{1/2}{e}_{j}}\right\rangle \right| = \left| \left\langle {{\rho }^{1/2}{e}_{j},{A}_{n}{\rho }^{1/2}{e}_{j}}\right\rangle \right| \leq C{\begin{Vmatrix}{\rho }^{1/2}{e}_{j}\end{Vmatrix}}^{2}, \] and, \[ \mathop{\sum }\limits_{j}{\begin{Vmatrix}{\rho }^{1/2}{e}_{j}\end{Vmatrix}}^{2} = \mathop{\sum }\limits_{j}\left\langle {{\rho }^{1/2}{e}_{j},{\rho }^{1/2}{e}_{j}}\right\rangle = \mathop{\sum }\limits_{j}\left\langle {{e}_{j},\rho {e}_{j}}\right\rangle = \operatorname{trace}\left( \rho \right) < \infty . \] Furthermore, since \( {A}_{n}\left( {{\rho }^{1/2}{e}_{j}}\right) \) converges to \( A\left( {{\rho }^{1/2}{e}_{j}}\right) \) for each \( j \), dominated convergence tells us that \[ \operatorname{trace}\left( {{\rho }^{1/2}A{\rho }^{1/2}}\right) = \mathop{\sum }\limits_{j}\left\langle {{e}_{j},{\rho }^{1/2}A{\rho }^{1/2}{e}_{j}}\right\rangle \] \[ = \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum }\limits_{j}\left\langle {{e}_{j},{\rho }^{1/2}{A}_{n}{\rho }^{1/2}{e}_{j}}\right\rangle \] \[ = \mathop{\lim }\limits_{{n \rightarrow \infty }}\operatorname{trace}\left( {{\rho }^{1/2}{A}_{n}{\rho }^{1/2}}\right) . \] As in (19.1), we can shift the second factor of \( {\rho }^{1/2} \) to the front of the trace to obtain Point 4 in Definition 19.6. ∎ | Yes |
Given a system of linear inequalities \( {Ax} \leq b \), formulate the problem of finding a solution to \( {Ax} \leq b \) as a problem of finding a solution to a system of linear inequalities in \( n - 1 \) variables using Fourier's elimination method. | Given a system of linear inequalities \( {Ax} \leq b \), let \( {A}^{n} \mathrel{\text{:=}} A,{b}^{n} \mathrel{\text{:=}} b \) ;
For \( i = n,\ldots ,1 \), eliminate variable \( {x}_{i} \) from \( {A}^{i}x \leq {b}^{i} \) with the above procedure to obtain system \( {A}^{i - 1}x \leq {b}^{i - 1} \) .
System \( {A}^{1}x \leq {b}^{1} \), which involves variable \( {x}_{1} \) only, is of the type, \( {x}_{1} \leq {b}_{p}^{1} \) , \( p \in P, - {x}_{1} \leq {b}_{q}^{1}, q \in N \), and \( 0 \leq {b}_{i}^{1}, i \in Z \) .
System \( {A}^{0}x \leq {b}^{0} \) has the following inequalities: \( 0 \leq {b}_{pq}^{0} \mathrel{\text{:=}} {b}_{p}^{1} + {b}_{q}^{1} \) , \( p \in P, q \in N,0 \leq {b}_{i}^{0} \mathrel{\text{:=}} {b}_{i}^{1}, i \in Z. \)
Applying Theorem 3.1, we obtain that \( {Ax} \leq b \) is feasible if and only if \( {A}^{0}x \leq {b}^{0} \) is feasible, and this happens when the \( {b}_{pq}^{0} \) and \( {b}_{i}^{0} \) are all nonnegative. | Yes |
The period 2 elliptic points of \( {\Gamma }_{0}\left( N\right) \) are in bijective correspondence with the ideals \( J \) of \( \mathbb{Z}\left\lbrack i\right\rbrack \) such that \( \mathbb{Z}\left\lbrack i\right\rbrack /J \cong \mathbb{Z}/N\mathbb{Z} \) . | This is an application of beginning algebraic number theory; see for example Chapter 9 of [IR92] for the results to quote. For period 2, the ring \( A = \mathbb{Z}\left\lbrack i\right\rbrack \) is a principal ideal domain and its maximal ideals are - for each prime \( p \equiv 1\left( {\;\operatorname{mod}\;4}\right) \), two ideals \( {J}_{p} = \left\langle {a + bi}\right\rangle \) and \( {\bar{J}}_{p} = \langle a + \) \( \left. {b{\bar{i}}}\right\rangle \) such that \( \langle p\rangle = {J}_{p}{\bar{J}}_{p} \) and the quotients \( A/{J}_{p}^{e} \) and \( A/{\bar{J}}_{p}^{e} \) are group-isomorphic to \( \mathbb{Z}/{p}^{e}\mathbb{Z} \) for all \( e \in \mathbb{N} \) , - for each prime \( p \equiv - 1\left( {\;\operatorname{mod}\;4}\right) \), the ideal \( {J}_{p} = \langle p\rangle \) such that the quotient \( A/{J}_{p}^{e} \) is group-isomorphic to \( {\left( \mathbb{Z}/{p}^{e}\mathbb{Z}\right) }^{2} \) for all \( e \in \mathbb{N} \) , - for \( p = 2 \), the ideal \( {J}_{2} = \left\langle {1 + i}\right\rangle \) such that \( \langle 2\rangle = {J}_{2}^{2} \) and the quotient \( A/{J}_{2}^{e} \) is group-isomorphic to \( {\left( \mathbb{Z}/{2}^{e}\mathbb{Z}\right) }^{2} \) for even \( e \in \mathbb{N} \) and is group-isomorphic to \( \mathbb{Z}/{2}^{\left( {e + 1}\right) /2}\mathbb{Z} \oplus \mathbb{Z}/{2}^{\left( {e - 1}\right) /2}\mathbb{Z} \) for odd \( e \in \mathbb{N} \). | No |
Theorem 13.31. \( {\mathcal{X}}_{\infty } \sim {\Lambda }^{{r}_{2}} \oplus \) ( \( \Lambda \) -torsion). | Null | No |
Theorem 2.8. For any compact subset \( M \) of \( {\mathbb{R}}^{d} \), the convex hull conv \( M \) is again compact. | Let \( {\left( {y}_{v}\right) }_{v \in \mathbb{N}} \) be any sequence of points from conv \( M \) . We shall prove that the sequence admits a subsequence which converges to a point in conv \( M \) . Let the dimension of aff \( M \) be denoted by \( n \) . Then Corollary 2.5 shows that each \( {y}_{v} \) in the sequence has a representation
\[
{y}_{v} = \mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{\lambda }_{vi}{x}_{vi}
\]
where \( {x}_{vi} \in M \) . We now consider the \( n + 1 \) sequences
\[
{\left( {x}_{v1}\right) }_{v \in \mathbb{N}},\ldots ,{\left( {x}_{v\left( {n + 1}\right) }\right) }_{v \in \mathbb{N}}
\]
(6)
of points from \( M \), and the \( n + 1 \) sequences
\[
{\left( {\lambda }_{v1}\right) }_{v \in \mathbb{N}},\ldots ,{\left( {\lambda }_{v\left( {n + 1}\right) }\right) }_{v \in \mathbb{N}}
\]
(7)
of real numbers from \( \left\lbrack {0,1}\right\rbrack \) . By the compactness of \( M \) there is a subsequence of \( {\left( {x}_{v1}\right) }_{v \in \mathbb{N}} \) which converges to a point in \( M \) . Replace all \( 2\left( {n + 1}\right) \) sequences by the corresponding subsequences. Change notation such that (6) and (7) now denote the subsequences; then \( {\left( {x}_{v1}\right) }_{v \in \mathbb{N}} \) converges in \( M \) . Next, use the compactness of \( M \) again to see that there is a subsequence of the (sub)sequence \( {\left( {x}_{v2}\right) }_{v \in \mathbb{N}} \) which converges to a point in \( M \) . Change notation, etc. Then after \( 2\left( {n + 1}\right) \) steps, where we use the compactness of \( M \) in step \( 1,\ldots, n + 1 \) , and the compactness of \( \left\lbrack {0,1}\right\rbrack \) in step \( n + 2,\ldots ,{2n} + 2 \), we end up with subsequences
\[
{\left( {x}_{{v}_{m}1}\right) }_{m \in \mathbb{N}},\ldots ,{\left( {x}_{{v}_{m}\left( {n + 1}\right) }\right) }_{m \in \mathbb{N}}
\]
of the original sequences (6) which converge in \( M \), say
\[
\mathop{\lim }\limits_{{m \rightarrow \infty }}{x}_{{v}_{m}i} = {x}_{0i},\;i = 1,\ldots, n + 1,
\]
and subsequences
\[
{\left( {\lambda }_{{v}_{m}1}\right) }_{m \in \mathbb{N}},\ldots ,{\left( {\lambda }_{{v}_{m}\left( {n + 1}\right) }\right) }_{m \in \mathbb{N}}
\]
of the original sequences (7) which converge in \( \left\lbrack {0,1}\right\rbrack \), say
\[
\mathop{\lim }\limits_{{m \rightarrow \infty }}{\lambda }_{{v}_{m}i} = {\lambda }_{0i},\;i = 1,\ldots, n + 1.
\]
Since
\[
\mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{\lambda }_{{v}_{m}i} = 1,\;m \in \mathbb{N}
\]
we also have
\[
\mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{\lambda }_{0i} = 1
\]
Then the linear combination
\[
{y}_{0} \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = 1}}^{{n + 1}}{\lambda }_{0i}{x}_{0i}
\]
is in fact a convex combination. Therefore, \( {y}_{0} \) is in conv \( M \) by Theorem 2.2. It is also clear that
\[
\mathop{\lim }\limits_{{m \rightarrow \infty }}{y}_{{v}_{m}} = {y}_{0}
\]
In conclusion, \( {\left( {y}_{{v}_{m}}\right) }_{m\; \in \;\mathbb{N}} \) is a subsequence of \( {\left( {y}_{v}\right) }_{v\; \in \;\mathbb{N}} \) which converges to a point in conv \( M \) . | Yes |
Theorem 4 (Lindenstrauss-Pelczynski). Let \( \left( {x}_{n}\right) \) be a normalized unconditional basis of \( {c}_{0} \) . Then \( \left( {x}_{n}\right) \) is equivalent to the unit vector basis. | Null | No |
Let \( \left( {M, F}\right) \) be a Finsler manifold. Suppose that at some \( p \in M \), the exponential map \( {\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering projection. Let \( {\sigma }_{0}\left( t\right) \mathrel{\text{:=}} {\exp }_{p}\left( {t{T}_{0}}\right) \) and \( {\sigma }_{1}\left( t\right) \mathrel{\text{:=}} {\exp }_{p}\left( {t{T}_{1}}\right) ,0 \leq t \leq L \) be any two (smooth) geodesics emanating from \( p \) and terminating at some common \( q \in M \). If \( {\sigma }_{0} \) is homotopic to \( {\sigma }_{1} \) through a homotopy with fixed endpoints \( p \) and \( q \), then \( {T}_{0} = {T}_{1} \) (equivalently, \( {\sigma }_{0} = {\sigma }_{1} \)). | The contrapositive of the first conclusion encompasses the second conclusion. So it suffices to establish the first one. Suppose \( {\sigma }_{0} \) is homotopic to \( {\sigma }_{1} \), through a homotopy \( h\left( {t, u}\right) ,0 \leq t \leq L \) , \( 0 \leq u \leq 1 \) with fixed endpoints \( p \) and \( q \) . Using Theorem 9.3.1, we lift this \( h \) to a homotopy \( \widetilde{h} : \left\lbrack {0, L}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow {T}_{p}M \) with \( \widetilde{h}\left( {t,0}\right) = t{T}_{0} \) . By hypothesis, every \( t \) -curve of the homotopy \( h \) begins at \( p \) and ends at \( q \) . Theorem 9.3.1 assures us that correspondingly, every \( t \) -curve of the lifted homotopy \( \widetilde{h} \) begins at the origin of \( {T}_{p}M \) and ends at the tip of \( L{T}_{0} \in {T}_{p}M \) . Note that both \( \widetilde{h}\left( {t,1}\right) \) and \( t{T}_{1} \) are lifts of \( {\sigma }_{1} \) which emanate from the origin of \( {T}_{p}M \) . So, by a corollary of Theorem 9.3.1, they must be the same. Consequently, \( \widetilde{h} \) is a homotopy between the rays \( t{T}_{0} \) and \( t{T}_{1} \), and all the intermediate \( t \) -curves share the same endpoints. However, the only way for the two rays \( t{T}_{0} \) and \( t{T}_{1},0 \leq t \leq L \), to have the same endpoints would be \( {T}_{0} = {T}_{1} \) . This is equivalent to saying that \( {\sigma }_{0} \) is actually identical to \( {\sigma }_{1} \). | Yes |
Let \( M \) be a compact connected oriented manifold of dimension \( {2n}, n \) odd. Then for \( G = \mathbb{Z} \) or any field \( \mathbb{F} \) of characteristic not equal to 2, \( \operatorname{rank}\left( {{K}^{n}\left( {M;G}\right) }\right) \) is even. Also, the Euler characteristic \( \chi \left( M\right) \) is even. | By Theorem 6.4.8, \( \langle \) , \( \rangle {isanonsingularskew} - {symmetricbilinearformon} \) \( {K}^{n}\left( {M;G}\right) \), so by Theorem B.2.1, \( {K}^{n}\left( {M;G}\right) \) must have even rank.
We may use any field to compute Euler characteristic. Choosing \( \mathbb{F} = \mathbb{Q} \), say, and using Poincaré duality, a short calculation shows
\[
\chi \left( M\right) = \mathop{\sum }\limits_{{k = 0}}^{{2n}}{\left( -1\right) }^{k}\dim {H}_{k}\left( {M;\mathbb{Q}}\right)
\]
\[
= 2\left( {\mathop{\sum }\limits_{{k = 0}}^{{n - 1}}{\left( -1\right) }^{k}\dim {H}_{k}\left( {M;\mathbb{Q}}\right) }\right) - \dim {H}_{n}\left( {M;\mathbb{Q}}\right)
\]
which is even. | Yes |
Proposition 12.22. Every continuous action of a compact topological group on a Hausdorff space is proper. | Suppose \( G \) is a compact group acting continuously on a Hausdorff space \( E \), and let \( \Theta : G \times E \rightarrow E \times E \) be the map defined by (12.5). Given a compact set \( L \subseteq E \times E \), let \( K = {\pi }_{2}\left( L\right) \), where \( {\pi }_{2} : E \times E \rightarrow E \) is the projection on the second factor. Because \( E \times E \) is Hausdorff, \( L \) is closed in \( E \times E \) . Thus \( {\Theta }^{-1}\left( L\right) \) is a closed subset of the compact set \( G \times K \), hence compact. | Yes |
Let \( {a}_{1},\ldots ,{a}_{n} \) for \( n \geq 2 \) be nonzero integers. Suppose there is a prime \( p \) and positive integer \( h \) such that \( {p}^{h} \mid {a}_{i} \) for some \( i \) and \( {p}^{h} \) does not divide \( {a}_{j} \) for all \( j \neq i \) . Then show that \( S = \frac{1}{{a}_{1}} + \cdots + \frac{1}{{a}_{n}} \) is not an integer. | Theorem 1.1.14 Given \( a, n \in \mathbb{Z},{a}^{\phi \left( n\right) } \equiv 1\left( {\;\operatorname{mod}\;n}\right) \) when \( \gcd \left( {a, n}\right) = 1 \). This is a theorem due to Euler.\n\nProof. The case where \( n \) is prime is clearly a special case of Fermat’s little Theorem. The argument is basically the same as that of the alternate solution to Exercise 1.1.13.\n\nConsider the ring \( \mathbb{Z}/n\mathbb{Z} \) . If \( a, n \) are coprime, then \( \bar{a} \) is a unit in this ring. The units form a multiplicative group of order \( \phi \left( n\right) \), and so clearly \( {\bar{a}}^{\phi \left( n\right) } = \overline{1} \) . Thus, \( {a}^{\phi \left( n\right) } \equiv 1\left( {\;\operatorname{mod}\;n}\right) \) . | No |
Let \( X \) be strongly regular with eigenvalues \( k > \theta > \tau \) . Suppose that \( x \) is an eigenvector of \( {A}_{1} \) with eigenvalue \( {\sigma }_{1} \) such that \( {\mathbf{1}}^{T}x = 0 \). If \( {Bx} = 0 \), then \( {\sigma }_{1} \in \{ \theta ,\tau \} \), and if \( {Bx} \neq 0 \), then \( \tau < {\sigma }_{1} < \theta \) . | Since \( {\mathbf{1}}^{T}x = 0 \), we have
\[
\left( {{A}_{1}^{2} - \left( {a - c}\right) {A}_{1} - \left( {k - c}\right) I}\right) x = - {B}^{T}{Bx}
\]
and since \( X \) is strongly regular with eigenvalues \( k,\theta \), and \( \tau \), we have
\[
\left( {{A}_{1}^{2} - \left( {a - c}\right) {A}_{1} - \left( {k - c}\right) I}\right) x = \left( {{A}_{1} - {\theta I}}\right) \left( {{A}_{1} - {\tau I}}\right) x.
\]
Therefore, if \( x \) is an eigenvector of \( {A}_{1} \) with eigenvalue \( {\sigma }_{1} \) ,
\[
\left( {{\sigma }_{1} - \theta }\right) \left( {{\sigma }_{1} - \tau }\right) x = - {B}^{T}{Bx}.
\]
If \( {Bx} = 0 \), then \( \left( {{\sigma }_{1} - \theta }\right) \left( {{\sigma }_{1} - \tau }\right) = 0 \) and \( {\sigma }_{1} \in \{ \theta ,\tau \} \) . If \( {Bx} \neq 0 \), then \( {B}^{T}{Bx} \neq 0 \), and so \( x \) is an eigenvector for the positive semidefinite matrix \( {B}^{T}B \) with eigenvalue \( - \left( {{\sigma }_{1} - \theta }\right) \left( {{\sigma }_{1} - \tau }\right) \) . It follows that \( \left( {{\sigma }_{1} - \theta }\right) \left( {{\sigma }_{1} - \tau }\right) < 0 \) , whence \( \tau < {\sigma }_{1} < \theta \) . | Yes |
Let \( A \in B\left( {{L}_{2}\left( {\mathbb{R}}^{m}\right) }\right) \) be a real \( {}^{3} \) positivity improving selfadjoint operator. Assume that \( \parallel A\parallel \) is an eigenvalue of \( A \) . Then the multiplicity of the eigenvalue \( \parallel A\parallel \) equals 1 and there is an \( f > 0 \) that spans the eigenspace \( N\left( {\parallel A\parallel - A}\right) \) . | Assume that \( f \neq 0 \) and \( {Af} = \parallel A\parallel f \) . Since \( A \) is real, we may assume that \( f \) is real (otherwise we could replace \( f \) by \( \operatorname{Re}f \) or \( \operatorname{Im}f \), because \( A\left( {\operatorname{Re}f}\right) = A\left( {f + {Kf}}\right) /2 = \left( {{Af} + {KAf}}\right) /2 = \left( {\parallel A\parallel f + K\parallel A\parallel f}\right) /2 = \) \( \parallel A\parallel \left( {\operatorname{Re}f}\right) \) and \( A\left( {\operatorname{Im}f}\right) = \parallel A\parallel \left( {\operatorname{Im}f}\right) ) \) . From the inequality \( \pm f \leq \left| f\right| \) it follows that \( \pm {Af} \leq A\left| f\right| \) . Therefore, \( \left| {Af}\right| \leq A\left| f\right| \), and thus
\[
\langle f,{Af}\rangle \leq \langle \left| f\right| ,\left| {Af}\right| \rangle \leq \langle \left| f\right|, A\left| f\right| \rangle .
\]
This implies that
\[
\parallel A\parallel \parallel f{\parallel }^{2} = \langle f,{Af}\rangle \leq \langle \left| f\right|, A\left| f\right| \rangle \leq \parallel A\parallel \parallel \left| f\right| {\parallel }^{2} = \parallel A\parallel \parallel f{\parallel }^{2},
\]
i.e., that
\[
\langle f,{Af}\rangle = \langle \left| f\right|, A\left| f\right| \rangle
\]
Let us define \( {f}_{ + } \) and \( {f}_{ - } \) by the equalities
\[
{f}_{ + }\left( x\right) = \max \{ 0, f\left( x\right) \} ,\;{f}_{ - } = {f}_{ + } - f.
\]
Then \( \left| f\right| = {f}_{ + } + {f}_{ - } \) . Consequently,
\[
\left\langle {{f}_{ + }, A{f}_{ - }}\right\rangle = \frac{1}{4}\{ \langle \left| f\right|, A\left| f\right| \rangle - \langle f,{Af}\rangle \} = 0.
\]
Hence we have \( {f}_{ + } = 0 \) or \( {f}_{ - } = 0 \), since \( {f}_{ + } \neq 0 \) and \( {f}_{ - } \neq 0 \) imply that \( A{f}_{ - } > 0 \), and thus that \( \left\langle {{f}_{ + }, A{f}_{ - }}\right\rangle \neq 0 \) . Consequently, we have proved that: \( f \geq 0 \) or \( f \leq 0 \) . We can assume, without loss or generality, that \( f \geq 0 \) . Since \( f = \parallel A{\parallel }^{-1}{Af} \) and \( f \neq 0 \), it then follows that we even have \( f > 0 \), because \( A \) is positivity improving.
The theorem will be proved if we show that \( f \) spans the space \( N(\parallel A\parallel - \) \( A) \) . For every element \( g \) of \( N\left( {\parallel A\parallel - A}\right) \) the functions \( \operatorname{Re}g \) and \( \operatorname{Im}g \) do not change sign. Such an element can only be orthogonal to the positive element \( f \) if \( g = 0 \) . Therefore, \( N\left( {\parallel A\parallel - A}\right) = L\left( f\right) \) . | Yes |
Theorem 8.21 METRIC TSP admits a polynomial-time 2-approximation algorithm. | Applying the Jarnik-Prim Algorithm (6.9), we first find a minimum-weight spanning tree \( T \) of \( G \) . Suppose that \( C \) is a minimum-weight Hamilton cycle of \( G \) . By deleting any edge of \( C \), we obtain a Hamilton path \( P \) of \( G \) . Because \( P \) is a spanning tree of \( G \) and \( T \) is a spanning tree of minimum weight,\[
w\left( T\right) \leq w\left( P\right) \leq w\left( C\right)
\]We now duplicate each edge of \( T \), thereby obtaining a connected even graph \( H \) with \( V\left( H\right) = V\left( G\right) \) and \( w\left( H\right) = {2w}\left( T\right) \) . Note that this graph \( H \) is not even a subgraph of \( G \), let alone a Hamilton cycle. The idea is to transform \( H \) into a Hamilton cycle of \( G \), and to do so without increasing its weight. More precisely, we construct a sequence \( {H}_{0},{H}_{1},\ldots ,{H}_{n - 2} \) of connected even graphs, each with vertex set \( V\left( G\right) \), such that \( {H}_{0} = H,{H}_{n - 2} \) is a Hamilton cycle of \( G \), and \( w\left( {H}_{i + 1}\right) \leq w\left( {H}_{i}\right) \) , \( 0 \leq i \leq n - 3 \) . We do so by reducing the number of edges, one at a time, as follows. | Yes |
Corollary 11.3.2. There exists a point \( x \in M \) such that \( G \cdot x \) is closed in \( M \). | Let \( y \in M \) and let \( Y \) be the closure of \( G \cdot y \). Then \( G \cdot y \) is open in \( Y\), by the argument in the proof of Theorem 11.3.1, and hence \( Z = Y - G \cdot y \) is closed in \( Y \) Thus \( Z \) is quasiprojective. Furthermore, \( \dim Z < \dim Y \) by Theorem A.1.19, and \( Z \) is a union of orbits. This implies that an orbit of minimal dimension is closed. | No |
Theorem 8.4. Let \( A \) be a finitely generated torsion-free abelian group. Then \( A \) is free. | Assume \( A \neq 0 \) . Let \( S \) be a finite set of generators, and let \( {x}_{1},\ldots ,{x}_{n} \) be a maximal subset of \( S \) having the property that whenever \( {v}_{1}{x}_{1} + \cdots + {v}_{n}{x}_{n} = 0 \) then \( {v}_{j} = 0 \) for all \( j \) . (Note that \( n \geqq 1 \) since \( A \neq 0 \) ). Let \( B \) be the subgroup generated by \( {x}_{1},\ldots ,{x}_{n} \) . Then \( B \) is free. Given \( y \in S \) there exist integers \( {m}_{1},\ldots ,{m}_{n}, m \) not all zero such that
\[
{my} + {m}_{1}{x}_{1} + \cdots + {m}_{n}{x}_{n} = 0,
\]
by the assumption of maximality on \( {x}_{1},\ldots ,{x}_{n} \) . Furthermore, \( m \neq 0 \) ; otherwise all \( {m}_{j} = 0 \) . Hence \( {my} \) lies in \( B \) . This is true for every one of a finite set of generators \( y \) of \( A \), whence there exists an integer \( m \neq 0 \) such that \( {mA} \subset B \) . The map
\[
x \mapsto {mx}
\]
of \( A \) into itself is a homomorphism, having trivial kernel since \( A \) is torsion free. Hence it is an isomorphism of \( A \) onto a subgroup of \( B \) . By Theorem 7.3 of the preceding section, we conclude that \( {mA} \) is free, whence \( A \) is free. | Yes |
Theorem 2.9. Schönflies Theorem. Let \( e : {S}^{2} \rightarrow {S}^{3} \) be any piecewise linear embedding. Then \( {S}^{3} - e{S}^{2} \) has two components, the closure of each of which is a piecewise linear ball. | No proof will be given here for this fundamental, non-trivial result (for a proof see [81]). The piecewise linear condition has to be inserted, as there exist the famous "wild horned spheres" that are are examples of topological embeddings \( e : {S}^{2} \rightarrow {S}^{3} \) for which the complementary components are not even simply connected. | No |
Suppose that \( X \) and \( Y \) are graphs with minimum valency four. Then \( X \cong Y \) if and only if \( L\left( X\right) \cong L\left( Y\right) \) . | Let \( C \) be a clique in \( L\left( X\right) \) containing exactly \( c \) vertices. If \( c > 3 \) , then the vertices of \( C \) correspond to a set of \( c \) edges in \( X \), meeting at a common vertex. Consequently, there is a bijection between the vertices of \( X \) and the maximal cliques of \( L\left( X\right) \) that takes adjacent vertices to pairs of cliques with a vertex in common. The remaining details are left as an exercise. | No |
An intersection of closed cells is a closed cell. | Corollary 1.26. An intersection of closed cells is a closed cell. | No |
The LAPLACE transform of a function \( f \) is defined to be another function \( \widetilde{f} \), given by \( \widetilde{f}\left( s\right) = {\int }_{0}^{\infty }f\left( t\right) {e}^{-{st}}{dt} \) for all \( s \) such that the integral is convergent. | With this definition one finds that \( \widetilde{\delta }\left( s\right) = 1 \) for all \( s \) . Similarly, \( {\widetilde{\delta }}_{a}\left( s\right) = {e}^{-{as}} \) , if \( a > 0 \) . | No |
Let \( \varphi : A \rightarrow B \) be a morphism in an additive category. Then \( \varphi \) is a monomorphism if and only if \( 0 \rightarrow A \) is its kernel, and \( \varphi \) is an epimorphism if and only if \( B \rightarrow 0 \) is its cokernel. | First assume \( \varphi : A \rightarrow B \) is a monomorphism. If \( \zeta : Z \rightarrow A \) is any morphism such that the composition \( Z \rightarrow A \rightarrow B \) is 0, then \( \zeta \) is 0 by Lemma 1.3, and in particular \( \zeta \) factors (uniquely) through \( 0 \rightarrow A \) . This proves that \( 0 \rightarrow A \) is a kernel of \( \varphi \), as stated.
Conversely, assume that \( 0 \rightarrow A \) is a kernel for \( \varphi : A \rightarrow B \), and let \( \zeta : Z \rightarrow A \) be a morphism such that \( \varphi \circ \zeta = 0 \) . It follows that \( \zeta \) factors through \( 0 \rightarrow A \), since the latter is a kernel for \( \varphi \) :
This implies \( \zeta = 0 \), proving that \( \varphi \) is a monomorphism. | Yes |
Theorem 6.94 (Sum rule for limiting directional subdifferentials). Let \( X \) be a WCG space, and let \( f = {f}_{1} + \cdots + {f}_{k} \), where \( {f}_{1},\ldots ,{f}_{k} \in \mathcal{L}\left( X\right) \) . Then\n\n{\partial }_{\ell }f\left( \bar{x}\right) \subset {\partial }_{\ell }{f}_{1}\left( \bar{x}\right) + \cdots + {\partial }_{\ell }{f}_{k}\left( \bar{x}\right) . | Proof. We know that \( X \) is H-smooth. Let \( {\bar{x}}^{ * } \in {\partial }_{\ell }f\left( \bar{x}\right) \), and let \( \left( {x}_{n}\right) \rightarrow \bar{x},\left( {x}_{n}^{ * }\right) \overset{ * }{ \rightarrow }{\bar{x}}^{ * } \) with \( {x}_{n}^{ * } \in {\partial }_{H}f\left( {x}_{n}\right) \) for all \( n \) . Given a weak* closed neighborhood \( V \) of 0 in \( {X}^{ * } \) and a sequence \( \left( {\varepsilon }_{n}\right) \rightarrow {0}_{ + } \), by the fuzzy sum rule for Hadamard subdifferentials (Theorem 4.69) there are sequences \( \left( \left( {{x}_{i, n},{x}_{i, n}^{ * }}\right) \right) \in {\partial }_{H}{f}_{i} \), for \( i \in {\mathbb{N}}_{k} \), such that \( d\left( {{x}_{i, n},{x}_{n}}\right) \leq {\varepsilon }_{n},{x}_{n}^{ * } \in {x}_{1, n}^{ * } + \cdots + {x}_{k, n}^{ * } + V \) . Since for some \( r > 0 \) one has \( {x}_{i, n}^{ * } \in r{B}_{{X}^{ * }} \) for all \( \left( {i, n}\right) \in {\mathbb{N}}_{k} \times \mathbb{N} \), one can find \( {y}_{i}^{ * } \in {\partial }_{\ell }{f}_{i}\left( \bar{x}\right) \) such that \( \left( {x}_{i, n}^{ * }\right) { \rightarrow }^{ * }{y}_{i}^{ * } \) for \( i \in {\mathbb{N}}_{k} \) and \( {\bar{x}}^{ * } \in {y}_{1}^{ * } + \cdots + {y}_{k}^{ * } + V \) . Since \( S \mathrel{\text{:=}} {\partial }_{\ell }{f}_{1}\left( \bar{x}\right) + \cdots + {\partial }_{\ell }{f}_{k}\left( \bar{x}\right) \) is weak* compact and \( {\bar{x}}^{ * } \in S + V \) for every weak \( {}^{ * } \) closed neighborhood \( V \) of 0, one gets \( {\bar{x}}^{ * } \in S \) . | Yes |
The closed convex hull of a totally bounded set in a complete locally convex linear topological space is compact. | Let \( K \) be such a set in such a space. By the preceding lemma, \( \operatorname{co}\left( K\right) \) is totally bounded. Hence \( \overline{\mathrm{{co}}}\left( K\right) \) is closed and totally bounded. Since the ambient space is complete, \( \overline{\mathrm{{co}}}\left( K\right) \) is complete and totally bounded. Hence, by Theorem 6, it is compact. | Yes |
For \( k \geq 1 \) we have \[
\mathop{\sum }\limits_{{m = 0}}^{N}\frac{1}{m + x} = - \psi \left( x\right) + \log \left( {N + x}\right) - \mathop{\sum }\limits_{{j = 1}}^{k}\frac{{B}_{j}}{j{\left( N + x\right) }^{j}} + {R}_{k}\left( {-1, x, N}\right) ,
\] | where
\[
{R}_{k}\left( {-1, x, N}\right) = {\int }_{N}^{\infty }\frac{{B}_{k}\left( {\{ t\} }\right) }{{\left( t + x\right) }^{k + 1}}{dt}
\]
and \( \left| {{R}_{k}\left( {-1, x, N}\right) }\right| \leq \left| {{B}_{k + 2}/\left( {\left( {k + 2}\right) {\left( N + x\right) }^{k + 2}}\right) }\right| \) when \( k \) is even. | Yes |
If \( M \) is a compact surface with non-empty boundary then each path component of \( \partial M \) is a circle, and if \( C \) is one of those circles then the space obtained from \( M \) by attaching a 2-cell using as attaching map an embedding \( {S}^{1} \rightarrow M \) whose image is \( C \) is again a surface. | Thus, by attaching finitely many 2-cells in this way \( M \) becomes a closed surface, i.e., becomes a \( {T}_{g} \) or a \( {U}_{h} \) . | No |
A universal algebra \( A \) of type \( T \) is isomorphic to a subdirect product of \( {\left( {A}_{i}\right) }_{i \in I} \) if and only if there exist surjective homomorphisms \( {\varphi }_{i} : A \rightarrow {A}_{i} \) such that \( \mathop{\bigcap }\limits_{{i \in I}}\ker {\varphi }_{i} \) is the equality on \( A \). | Let \( P \) be a subdirect product of \( {\left( {A}_{i}\right) }_{i \in I} \) . The inclusion homomorphism \( \iota : P \rightarrow \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) and projections \( {\pi }_{j} : \mathop{\prod }\limits_{{i \in I}}{A}_{i} \rightarrow {A}_{j} \) yield surjective homomorphisms \( {\rho }_{i} = {\pi }_{i} \circ \iota : P \rightarrow {A}_{i} \) that separate the elements of \( P \), since elements of the product that have the same components must be equal. If now \( \theta : A \rightarrow P \) is an isomorphism, then the homomorphisms \( {\varphi }_{i} = {\rho }_{i} \circ \theta \) are surjective and separate the elements of \( A \) .
Conversely, assume that there exist surjective homomorphisms \( {\varphi }_{i} : A \rightarrow {A}_{i} \) that separate the elements of \( A \) . Then \( \varphi : x \mapsto {\left( {\varphi }_{i}\left( x\right) \right) }_{i \in I} \) is an injective homomorphism of \( A \) into \( \mathop{\prod }\limits_{{i \in I}}{A}_{i} \) . Hence \( A \cong \operatorname{Im}\varphi \) ; moreover, \( \operatorname{Im}\varphi \) is a subdirect product of \( {\left( {A}_{i}\right) }_{i \in I} \), since \( {\pi }_{i}\left( {\operatorname{Im}\varphi }\right) = {\varphi }_{i}\left( A\right) = {A}_{i} \) for all \( i.▱ \) | Yes |
If \( w \in {\Lambda }^{k}\left( E\right) \) and \( z \in {\Lambda }^{\ell }\left( E\right) \), then \( z \land w = {\left( -1\right) }^{k\ell }w \land z \). | Proposition 4.2 If \( w \in {\Lambda }^{k}\left( E\right) \) and \( z \in {\Lambda }^{\ell }\left( E\right) \), then \( z \land w = {\left( -1\right) }^{k\ell }w \land z.\) | Yes |
Theorem 2.7 Gaussian elimination for the simultaneous solution of an \( n \times n \) system for \( r \) different right-hand sides requires a total of | The computational cost, counting only the multiplications, in Gaussian elimination is \( {n}^{3}/3 + O\left( {n}^{2}\right) \) . It is left to the reader to show that the number of additions is also \( {n}^{3}/3 + O\left( {n}^{2}\right) \) (see Problem 2.7). | No |
Lemma 12.2. \( T : X \rightarrow Y \) is closed if and only if the following holds: When \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence in \( D\left( T\right) \) with \( {x}_{n} \rightarrow x \) in \( X \) and \( T{x}_{n} \rightarrow y \) in \( Y \), then \( x \in D\left( T\right) \) with \( y = {Tx} \) . | The closed graph theorem (recalled in Appendix B, Theorem B.16) implies that if \( T : X \rightarrow Y \) is closed and has \( D\left( T\right) = X \), then \( T \) is bounded. Thus for closed, densely defined operators, \( D\left( T\right) \neq X \) is equivalent with unboundedness. | No |
Theorem 3.14 (Gluing distributions together). Let \( {\left( {\omega }_{\lambda }\right) }_{\lambda \in \Lambda } \) be an arbitrary system of open sets in \( {\mathbb{R}}^{n} \) and let \( \Omega = \mathop{\bigcup }\limits_{{\lambda \in \Lambda }}{\omega }_{\lambda } \) . Assume that there is given a system of distributions \( {u}_{\lambda } \in {\mathcal{D}}^{\prime }\left( {\omega }_{\lambda }\right) \) with the property that \( {u}_{\lambda } \) equals \( {u}_{\mu } \) on \( {\omega }_{\lambda } \cap {\omega }_{\mu } \), for each pair of indices \( \lambda ,\mu \in \Lambda \) . Then there exists one and only one distribution \( u \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) such that \( {\left. u\right| }_{{\omega }_{\lambda }} = {u}_{\lambda } \) for all \( \lambda \in \Lambda \) . | Observe to begin with that there is at most one solution \( u \) . Namely, if \( u \) and \( v \) are solutions, then \( {\left. \left( u - v\right) \right| }_{{\omega }_{\lambda }} = 0 \) for all \( \lambda \) . This implies that \( u - v = 0 \), by Lemma 3.11.
We construct \( u \) as follows: Let \( {\left( {K}_{l}\right) }_{l \in \mathbb{N}} \) be a sequence of compact sets as in (2.4) and consider a fixed \( l \) . Since \( {K}_{l} \) is compact, it is covered by a finite subfamily \( {\left( {\Omega }_{j}\right) }_{j = 1,\ldots, N} \) of the sets \( {\left( {\omega }_{\lambda }\right) }_{\lambda \in \Lambda } \) ; we denote \( {u}_{j} \) the associated distributions given in \( {\mathcal{D}}^{\prime }\left( {\Omega }_{j}\right) \), respectively. By Theorem 2.17 there is a partition of unity \( {\psi }_{1},\ldots ,{\psi }_{N} \) consisting of functions \( {\psi }_{j} \in {C}_{0}^{\infty }\left( {\Omega }_{j}\right) \) satisfying \( {\psi }_{1} + \cdots + {\psi }_{N} = 1 \) on \( {K}_{l} \) . For \( \varphi \in {C}_{{K}_{l}}^{\infty }\left( \Omega \right) \) we set
\[
\langle u,\varphi {\rangle }_{\Omega } = {\left\langle u,\mathop{\sum }\limits_{{j = 1}}^{N}{\psi }_{j}\varphi \right\rangle }_{\Omega } = \mathop{\sum }\limits_{{j = 1}}^{N}{\left\langle {u}_{j},{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j}}.
\]
(3.39)
In this way, we have given \( \langle u,\varphi \rangle \) a value which apparently depends on a lot of choices (of \( l \), of the subfamily \( {\left( {\Omega }_{j}\right) }_{j = 1,\ldots, N} \) and of the partition of unity \( \left. \left\{ {\psi }_{j}\right\} \right) \) . But if \( {\left( {\Omega }_{k}^{\prime }\right) }_{k = 1,\ldots, M} \) is another subfamily covering \( {K}_{l} \), and \( {\psi }_{1}^{\prime },\ldots ,{\psi }_{M}^{\prime } \) is an associated partition of unity, we have, with \( {u}_{k}^{\prime } \) denoting the distribution given on \( {\Omega }_{k}^{\prime } \) :
\[
\mathop{\sum }\limits_{{j = 1}}^{N}{\left\langle {u}_{j},{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j}} = \mathop{\sum }\limits_{{j = 1}}^{N}\mathop{\sum }\limits_{{k = 1}}^{M}{\left\langle {u}_{j},{\psi }_{k}^{\prime }{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j}} = \mathop{\sum }\limits_{{j = 1}}^{N}\mathop{\sum }\limits_{{k = 1}}^{M}{\left\langle {u}_{j},{\psi }_{k}^{\prime }{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j} \cap {\Omega }_{k}^{\prime }}
\]
\[
= \mathop{\sum }\limits_{{j = 1}}^{N}\mathop{\sum }\limits_{{k = 1}}^{M}{\left\langle {u}_{k}^{\prime },{\psi }_{k}^{\prime }{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{k}^{\prime }} = \mathop{\sum }\limits_{{k = 1}}^{M}{\left\langle {u}_{k}^{\prime },{\psi }_{k}^{\prime }\varphi \right\rangle }_{{\Omega }_{k}^{\prime }},
\]
since \( {u}_{j} = {u}_{k}^{\prime } \) on \( {\Omega }_{j} \cap {\Omega }_{k}^{\prime } \) . This shows that \( u \) has been defined for \( \varphi \in \) \( {C}_{{K}_{l}}^{\infty }\left( \Omega \right) \) independently of the choice of finite subcovering of \( {K}_{l} \) and associated partition of unity. If we use such a definition for each \( {K}_{l}, l = 1,2,\ldots \), we find moreover that these definitions are consistent with each other. Indeed, for both \( {K}_{l} \) and \( {K}_{l + 1} \) one can use one cover and partition of unity chosen for \( {K}_{l + 1} \) . (In a similar way one finds that \( u \) does not depend on the choice of the sequence \( {\left( {K}_{l}\right) }_{l \in \mathbb{N}} \) .) This defines \( u \) as an element of \( {\mathcal{D}}^{\prime }\left( \Omega \right) \) .
Now we check the consistency of \( u \) with each \( {u}_{\lambda } \) as follows: Let \( \lambda \in \Lambda \) . For each \( \varphi \in {C}_{0}^{\infty }\left( {\omega }_{\lambda }\right) \) there is an \( l \) such that \( \varphi \in {C}_{{K}_{l}}^{\infty }\left( \Omega \right) \) . Then \( \langle u,\varphi \rangle \) can be defined by (3.39). Here
\[
\langle u,\varphi {\rangle }_{\Omega } = {\left\langle u,\mathop{\sum }\limits_{{j = 1}}^{N}{\psi }_{j}\varphi \right\rangle }_{\Omega } = \mathop{\sum }\limits_{{j = 1}}^{N}{\left\langle {u}_{j},{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j}}
\]
\[
= \mathop{\sum }\limits_{{j = 1}}^{N}{\left\langle {u}_{j},{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j} \cap {\omega }_{\lambda }} = \mathop{\sum }\limits_{{j = 1}}^{N}{\left\langle {u}_{\lambda },{\psi }_{j}\varphi \right\rangle }_{{\Omega }_{j} \cap {\omega }_{\lambda }} = {\left\langle {u}_{\lambda },\varphi \right\rangle }_{{\omega }_{\lambda }},
\]
which shows that \( {\left. u\right| }_{{\omega }_{\lambda }} = {u}_{\lambda } \) . | Yes |
Let \( {S}_{1} \) and \( {S}_{2} \) be any two projective subspaces of \( {\mathbb{P}}^{n}\left( k\right) \) . Then | Any subspace \( {k}^{r + 1} \) has codimension \( n - r \) in \( {k}^{n + 1} \) ; therefore the associated subspace \( {\mathbb{P}}^{r}\left( k\right) \) has the same codimension \( n - r \) in \( {\mathbb{P}}^{n}\left( k\right) \) . Then apply the corresponding vector space theorem. | No |
Theorem 2.1.4 (Greene-Krantz [GRK2]). Let \( B \subseteq {\mathbb{C}}^{n} \) be the unit ball. Let \( {\rho }_{0}\left( z\right) = {\left| z\right| }^{2} - 1 \) be the usual defining function for \( B \) . If \( \epsilon > 0 \) is sufficiently small, \( k = k\left( n\right) \) is sufficiently large, and \( \Omega \in {\mathcal{U}}_{\epsilon }^{k}\left( B\right) \) then either | either \[ \Omega \sim B \] (2.1.4.1) or \( \Omega \) is not biholomorphic to the ball and (2.1.4.2) (a) Aut \( \left( \Omega \right) \) is compact. (b) Aut \( \left( \Omega \right) \) has a fixed point. Moreover, If \( K \subset \subset B,\epsilon > 0 \) is sufficiently small (depending on \( K \) ), and \( \Omega \in {\mathcal{U}}_{\epsilon }^{k}\left( B\right) \) has the property that its fixed point set lies in \( K \), then there is a biholomorphic mapping \( \Phi : \Omega \rightarrow \Phi \left( \Omega \right) \equiv {\Omega }^{\prime } \subseteq {\mathbb{C}}^{n} \) such that \( \operatorname{Aut}\left( {\Omega }^{\prime }\right) \) is the restriction to \( {\Omega }^{\prime } \) of a subgroup of the group of unitary matrices. | Yes |
Theorem 5.3. Let \( \mathrm{S} \) be an extension ring of \( \mathrm{R} \) and \( \mathrm{s} \in \mathrm{S} \) . Then the following conditions are equivalent. | SKETCH OF PROOF. (i) \( \Rightarrow \) (ii) Suppose \( s \) is a root of the monic polynomial \( {f\varepsilon R}\left\lbrack x\right\rbrack \) of degree \( n \) . We claim that \( {1}_{R} = {s}^{0}, s,{s}^{2},\ldots ,{s}^{n - 1} \) generate \( R\left\lbrack s\right\rbrack \) as an \( R \) -module. As observed above, every element of \( R\left\lbrack s\right\rbrack \) is of the form \( g\left( s\right) \) for some \( g \in R\left\lbrack x\right\rbrack \) . By the Division Algorithm III.6.2 \( g\left( x\right) = f\left( x\right) q\left( x\right) + r\left( x\right) \) with \( \deg r < \deg f \) . Therefore in \( S, g\left( s\right) = f\left( s\right) q\left( s\right) + r\left( s\right) = 0 + r\left( s\right) = r\left( s\right) \) . Hence \( g\left( s\right) \) is an \( R \) -linear combination of \( {1}_{R}, s,{s}^{2},\ldots ,{s}^{m} \) with \( m = \deg r < \deg f = n \) .
(ii) \( \Rightarrow \) (iii) Let \( T = R\left\lbrack s\right\rbrack \) .
(iii) \( \Rightarrow \) (iv) Let \( B \) be the subring \( T \) . Since \( R \subset R\left\lbrack s\right\rbrack \subset T, B \) is an \( R\left\lbrack s\right\rbrack \) -module that is finitely generated as an \( R \) -module by (iii). Since \( {1}_{S} \in B,{uB} = 0 \) for any \( {u\varepsilon S} \) implies \( u = {u1s} = 0 \) ; that is, the annihilator of \( B \) in \( R\left\lbrack s\right\rbrack \) is 0 .
(iv) \( \Rightarrow \) (i) Let \( B \) be generated over \( R \) by \( {b}_{1},\ldots ,{b}_{n} \) . Since \( B \) is an \( R\left\lbrack s\right\rbrack \) -module \( s{b}_{i} \in B \) for each \( i \) . Therefore there exist \( {r}_{ij} \in R \) such that
\[
s{b}_{1} = {r}_{11}{b}_{1} + {r}_{12}{b}_{2} + \cdots + {r}_{1n}{b}_{n}
\]
\[
s{b}_{2} = {r}_{21}{b}_{1} + {r}_{22}{b}_{2} + \cdots + {r}_{2n}{b}_{n}
\]
\[
s{b}_{n} = {r}_{n1}{b}_{1} + {r}_{n2}{b}_{2} + \cdots + {r}_{nn}{b}_{n}.
\]
Consequently,
\[
\left( {{r}_{11} - s}\right) {b}_{1} + {r}_{12},{b}_{2} + \cdots + {r}_{1n}{b}_{n} = 0
\]
\[
{r}_{21}{b}_{1} + \left( {{r}_{22} - s}\right) {b}_{2} + \cdots + {r}_{2n}{b}_{n} = 0
\]
\[
\text{.}
\]
\[
\text{.}
\]
\[
{r}_{n1}{b}_{1} + {r}_{n2},{b}_{2} + \cdots + \left( {{r}_{nn} - s}\right) {b}_{n} = 0.
\]
Let \( M \) be the \( n \times n \) matrix \( \left( {r}_{ij}\right) \) and let \( d \in R\left\lbrack s\right\rbrack \) be the determinant of the matrix \( M - s{I}_{n} \) . Then \( d{b}_{i} = 0 \) for all \( i \) by Exercise VII.3.8. Since \( B \) is generated by the \( {b}_{i},{dB} = 0 \) . Since the annihilator of \( B \) in \( R\left\lbrack s\right\rbrack \) is zero by (iv) we must have \( d = 0 \) . If \( f \) is the polynomial \( \left| {M - x{I}_{n}}\right| \) in \( R\left\lbrack x\right\rbrack \), then one of \( f, - f \) is monic and
\[
\pm f\left( s\right) = \pm \left| {M - s{I}_{n}}\right| = \pm d = 0.
\]
Therefore \( s \) is integral over \( R \) . | No |
For all \( k,{\dim }_{K}{\widetilde{H}}_{k}\left( {\Delta ;K}\right) \leq {\dim }_{K}{\widetilde{H}}_{k}\left( {\Gamma ;K}\right) \) . | By considering an extension field of \( K \) if necessary, we may assume that \( K \) is infinite. Let \( {\Delta }^{e} \) denote the exterior algebraic shifted complex of \( \Delta \) . By Proposition 11.4.7 we have \( {\widetilde{H}}_{k}\left( {\Delta ;K}\right) \cong {\widetilde{H}}_{k}\left( {{\Delta }^{e};K}\right) \) . Thus we need to show that \( {\dim }_{K}{\widetilde{H}}_{k}\left( {{\Delta }^{e};K}\right) \leq {\dim }_{K}{\widetilde{H}}_{k}\left( {{\Gamma }^{e};K}\right) \) for all \( k \) . By using (11.6) one has \( {\beta }_{in}\left( {I}_{\Delta }\right) = {\dim }_{K}{\widetilde{H}}_{n - i - 2}\left( {\Delta ;K}\right) \) . Hence it remains to show that \( {\beta }_{in}\left( {I}_{{\Delta }^{e}}\right) \leq {\beta }_{in}\left( {I}_{{\Gamma }^{e}}\right) \) for all \( i \) . Inequality (11.5) says that \( {m}_{ \leq i}\left( {{J}_{{\Delta }^{e}}, j}\right) \geq {m}_{ \leq i}\left( {{J}_{{\Gamma }^{e}}, j}\right) \) for all \( i \) and \( j \) . It then follows from Corollary 11.3.9 that \( {\beta }_{{ii} + j}\left( {I}_{{\Delta }^{e}}\right) \leq {\beta }_{{ii} + j}\left( {I}_{{\Gamma }^{e}}\right) \) for all \( i \) and \( j \) . Thus in particular \( {\beta }_{\text{in }}\left( {I}_{{\Delta }^{e}}\right) \leq {\beta }_{\text{in }}\left( {I}_{{\Gamma }^{e}}\right) \) for all \( i \). | Yes |
Theorem 7. For spaces, connectivity is preserved by surjective mappings. That is, if \( \left\lbrack {X,\mathcal{O}}\right\rbrack \) is connected, and \( f : X \rightarrow Y \) is a mapping, then \( \left\lbrack {Y,{\mathcal{O}}^{\prime }}\right\rbrack \) is connected. | Suppose not. Then \( Y = U \cup V \), where \( U \) and \( V \) are disjoint, open, and nonempty. Therefore \( X = {f}^{-1}\left( U\right) \cup {f}^{-1}\left( V\right) \), and the latter sets are disjoint, open, and nonempty, which is impossible. | Yes |
A vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies the system (3.2) if and only if there exists \( {\bar{x}}_{n} \) such that \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1},{\bar{x}}_{n}}\right) \) satisfies \( {Ax} \leq b \) . | We already remarked the "if" statement. For the converse, assume there is a vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfying (3.2). Note that the first set of inequalities in (3.2) can be rewritten as \[ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{kj}^{\prime }{x}_{j} - {b}_{k}^{\prime } \leq {b}_{i}^{\prime } - \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}^{\prime }{x}_{j},\;i \in {I}^{ + }, k \in {I}^{ - }. \] (3.3) Let \( l : = \mathop{\max }\limits_{{k \in {I}^{ - }}}\{ \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{kj}^{\prime }{\bar{x}}_{j} - {b}_{k}^{\prime }\} \) and \( u : = \mathop{\min }\limits_{{i \in {I}^{ + }}}\{ {b}_{i}^{\prime } - \mathop{\sum }\limits_{{j = 1}}^{{n - 1}}{a}_{ij}^{\prime }{\bar{x}}_{j}\} , \) where we define \( l \mathrel{\text{:=}} - \infty \) if \( {I}^{ - } = \varnothing \) and \( u \mathrel{\text{:=}} + \infty \) if \( {I}^{ + } = \varnothing \) . Since \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n - 1}}\right) \) satisfies (3.3), we have that \( l \leq u \) . Therefore, for any \( {\bar{x}}_{n} \) such that \( l \leq {\bar{x}}_{n} \leq u \), the vector \( \left( {{\bar{x}}_{1},\ldots ,{\bar{x}}_{n}}\right) \) satisfies the system (3.1), which is equivalent to \( {Ax} \leq b \) . | Yes |
Corollary 4.34. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( Y \) is barreled. Then any linear map \( T \) from \( X \) onto \( Y \) is nearly open. | Use \( {\mathcal{B}}_{0} = \) all convex, balanced neighborhoods of 0 in \( X \) . If \( B \in {\mathcal{B}}_{0} \), and \( x \in X \), then \( x \in {cB} \Rightarrow T\left( x\right) \in T\left( {cB}\right) = {cT}\left( B\right) \), so \( T\left( B\right) \) is convex, balanced, and absorbent (since \( T \) is onto). Hence \( T{\left( B\right) }^{ - } \) is closed, convex (Proposition 2.13), balanced (Proposition 2.5), and absorbent, that is \( T{\left( B\right) }^{ - } \) is a barrel in \( Y \) . Since \( Y \) is assumed to be barreled, \( T{\left( B\right) }^{ - } \) is a neighborhood of 0 . Hence \( T \) is nearly open by Corollary 4.33. | Yes |
If \( \left( {K,\mu ;\varphi }\right) \) is a faithful topological measure-preserving system, then \( \varphi \left( K\right) = K \), i.e., \( \left( {K;\varphi }\right) \) is a surjective topological system. | Theorem 10.2 of Krylov and Bogoljubov tells that every topological system \( \left( {K;\varphi }\right) \) has at least one invariant probability measure, and hence gives rise to at least one topological measure-preserving system. (By the lemma above, this topological measure-preserving system cannot be faithful if \( \left( {K;\varphi }\right) \) is not a surjective system. But even if the topological system is surjective and uniquely ergodic, the arising measure-preserving system need not be faithful as Exercise 9 shows.) | No |
Proposition 16.45 (The Riemannian Density). Let \( \left( {M, g}\right) \) be a Riemannian manifold with or without boundary. There is a unique smooth positive density \( {\mu }_{g} \) on \( M \) , called the Riemannian density, with the property that | Uniqueness is immediate, because any two densities that agree on a basis must be equal. Given any point \( p \in M \), let \( U \) be a connected smooth coordinate neighborhood of \( p \). Since \( U \) is diffeomorphic to an open subset of Euclidean space, it is orientable. Any choice of orientation of \( U \) uniquely determines a Riemannian volume form \( {\omega }_{g} \) on \( U \), with the property that \( {\omega }_{g}\left( {{E}_{1},\ldots ,{E}_{n}}\right) = 1 \) for any oriented orthonormal frame. If we put \( {\mu }_{g} = \left| {\omega }_{g}\right| \), it follows easily that \( {\mu }_{g} \) is a smooth positive density on \( U \) satisfying (16.20). If \( U \) and \( V \) are two overlapping smooth coordinate neighborhoods, the two definitions of \( {\mu }_{g} \) agree where they overlap by uniqueness, so this defines \( {\mu }_{g} \) globally. | Yes |
Assuming the \( {ABC} \) Conjecture, show that there are infinitely many primes \( p \) such that \( {2}^{p - 1} ≢ 1\left( {\;\operatorname{mod}\;{p}^{2}}\right) \) . | Null | No |
Proposition 9.32 For each \( j = 1,2,\ldots, n \), define a domain \( \operatorname{Dom}\left( {P}_{j}\right) \subset \) \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) as follows: | Proof of Proposition 9.32. By Proposition 9.30, the operator of multiplication by \( {k}_{j} \) is an unbounded self-adjoint operator on \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \), with domain equal to the set of \( \phi \) for which \( {k}_{j}\phi \left( \mathbf{k}\right) \) belongs to \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \). It then follows from the unitarity of the Fourier transform that \( {P}_{j} = \hslash {\mathcal{F}}^{-1}{M}_{{k}_{j}}\mathcal{F} \) is self-adjoint on \( {\mathcal{F}}^{-1}\left( {\operatorname{Dom}\left( {M}_{{k}_{j}}\right) }\right) \), where \( {M}_{{k}_{j}} \) denotes multiplication by \( {k}_{j} \). The second characterization of \( \operatorname{Dom}\left( {P}_{j}\right) \) follows from Lemma 9.33. ∎ | Yes |
Proposition 7.4 Suppose \( \mathbf{A} \) and \( {\mathbf{A}}^{\prime } \) are two additive categories, and suppose \( \mathbf{A} \) contains a biproduct of any two objects. Suppose \( F : \mathbf{A} \rightarrow {\mathbf{A}}^{\prime } \) is a covariant functor. Then the following are equivalent. | Proof: (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) \( \Rightarrow \) (i) works the same way here as it did in Proposition 6.1. The technical point-that \( F\left( {\pi }_{1}\right) \) and \( F\left( {\pi }_{2}\right) \) are the \( {\pi }_{1}^{\prime } \) and \( {\pi }_{2}^{\prime } \) for which \( \left( {F\left( {A \oplus A}\right) ;F\left( {\varphi }_{1}\right), F\left( {\varphi }_{2}\right) ,{\pi }_{1}^{\prime },{\pi }_{2}^{\prime }}\right) \) is a biproduct in \( {\mathbf{A}}^{\prime } \) -is even the same. To see this, we must establish that \( F\left( {\pi }_{1}\right) \) and \( F\left( {\pi }_{2}\right) \) are fillers for the appropriate diagrams in the proof of Proposition 7.2. That is, we must check that \( {i}_{F\left( A\right) } = F\left( {\pi }_{1}\right) F\left( {\varphi }_{1}\right) = F\left( {\pi }_{2}\right) F\left( {\varphi }_{2}\right) \), while \( 0 = \) \( F\left( {\pi }_{1}\right) F\left( {\varphi }_{2}\right) = F\left( {\pi }_{2}\right) F\left( {\varphi }_{1}\right) \) . But \( {i}_{F\left( A\right) } = F\left( {i}_{A}\right) = F\left( {{\pi }_{1}{\varphi }_{1}}\right) = F\left( {\pi }_{1}\right) F\left( {\varphi }_{1}\right) \) ; similarly, \( {i}_{F\left( A\right) } = F\left( {\pi }_{2}\right) F\left( {\varphi }_{2}\right) \) . Also, \( F\left( {\pi }_{1}\right) F\left( {\varphi }_{2}\right) = F\left( {{\pi }_{1}{\varphi }_{2}}\right) = F\left( 0\right) \), and similarly \( F\left( {\pi }_{2}\right) F\left( {\varphi }_{1}\right) = F\left( 0\right) \), so it suffices to show that \( F\left( 0\right) = 0 \), that is, \( F \) (zero morphism) \( = \) zero morphism. Since the zero morphism is precisely the morphism which factors through "the" zero object (both in \( \mathbf{A} \) and \( {\mathbf{A}}^{\prime } \) ), it suffices to show that \( F \) (zero object) \( = \) zero object.
Let \( O \) denote a zero object of \( \mathbf{A} \) . Note that \( \left( {O;i, i, i, i}\right) \) is a biproduct of \( O \) with \( O \) in \( \mathbf{A} \), where \( i = {i}_{O} \) is the only element of \( \operatorname{Hom}\left( {O, O}\right) \) . Hence
\[
O\overset{i}{ \rightarrow }O\overset{i}{ \leftarrow }O
\]
is a coproduct in \( \mathbf{A} \), so
\[
F\left( O\right) \overset{F\left( i\right) }{ \rightarrow }F\left( O\right) \overset{F\left( i\right) }{ \leftarrow }F\left( O\right)
\]
is a coproduct in \( {\mathbf{A}}^{\prime } \) . By Proposition 7.2, there exist unique \( {\pi }_{1},{\pi }_{2} \in \) \( \operatorname{Hom}\left( {F\left( O\right), F\left( O\right) }\right) \) such that \( \left( {F\left( O\right) ;F\left( i\right), F\left( i\right) ,{\pi }_{1},{\pi }_{2}}\right) \) is a biproduct. Letting \( F\left( i\right) \) play the role of \( {\varphi }_{1},{i}_{F\left( O\right) } = {\pi }_{1}F\left( i\right) \) . Letting \( F\left( i\right) \) play the role of \( {\varphi }_{2},{\pi }_{1}F\left( i\right) = 0 \) . Hence \( {i}_{F\left( O\right) } = 0 \), so \( F\left( O\right) \) is a zero object. (See Exercise 4.) | Yes |
Theorem 12.6.2 Let \( \mathcal{Q} \) be the incidence structure whose points are the vectors of \( {C}^{ * } \), and whose lines are triples of mutually orthogonal vectors. Then either \( \mathcal{Q} \) has no lines, or \( \mathcal{Q} \) is a generalized quadrangle, possibly degenerate, with lines of size three. | Proof. A generalized quadrangle has the property that given any line \( \ell \) and a point \( P \) off that line, there is a unique point on \( \ell \) collinear with \( P \) . We show that \( \mathcal{Q} \) satisfies this axiom.
Suppose that \( x, y \), and \( a - b - x - y \) are the three points of a line of \( \mathcal{Q} \), and let \( z \) be an arbitrary vector in \( {C}^{ * } \), not equal to any of these three. Then
\[
\langle z, x\rangle + \langle z, y\rangle + \langle z, a - b - x - y\rangle = \langle z, a - b\rangle = 2.
\]
Since each of the three terms is either 0 or 1 , it follows that there is a unique term equal to 0, and hence \( z \) is collinear with exactly one of the three points of the line.
Therefore, \( \mathcal{Q} \) is a generalized quadrangle with lines of size three. | Yes |
An operator \( T \) from a separable Hilbert space \( H \) into \( {L}_{2}\left( M\right) \) is a Carleman operator if and only if \( T{f}_{n}\left( x\right) \rightarrow 0 \) almost everywhere in \( M \) for every null-sequence \( \left( {f}_{n}\right) \) from \( D\left( T\right) \) . | It is evident from the definition that every Carleman operator has this property. It remains to prove the reverse direction. By Theorem 6.15 it is sufficient to show that the series \( {\sum }_{n}{\left| T{e}_{n}\left( x\right) \right| }^{2} \) is almost everywhere convergent for every ONS \( \left\{ {{e}_{1},{e}_{2},\ldots }\right\} \) from \( D\left( T\right) \) . | Yes |
Corollary 3.3.6. Let \( G \) be a finite group. Then \( G \) is reductive. | Null | No |
Let \( \Gamma \) be a gallery of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{d}}\right) \) . If \( \Gamma \) is not minimal, then there is a gallery \( {\Gamma }^{\prime } \) with the same extremities as \( \Gamma \) such that \( {\Gamma }^{\prime } \) has type \( {\mathbf{s}}^{\prime } = \left( {{s}_{1},\ldots ,{\widehat{s}}_{i},\ldots ,{\widehat{s}}_{j},\ldots ,{s}_{d}}\right) \) for some \( i < j \) . | Proof. Since \( \Gamma \) is not minimal, Lemma 3.69 implies that the number of walls separating \( {C}_{0} \) from \( {C}_{d} \) is less than \( d \) . Hence the walls crossed by \( \Gamma \) cannot all be distinct; for if a wall is crossed exactly once by \( \Gamma \), then it certainly separates \( {C}_{0} \) from \( {C}_{d} \) . We can therefore find a root \( \alpha \) and indices \( i, j \), with \( 1 \leq i < j \leq d \), such that \( {C}_{i - 1} \) and \( {C}_{j} \) are in \( \alpha \) but \( {C}_{k} \in - \alpha \) for \( i \leq k < j \) ; see Figure 3.6. Let \( \phi \) be the folding with image \( \alpha \) . If we modify \( \Gamma \) by applying \( \phi \) to the portion \( {C}_{i},\ldots ,{C}_{j - 1} \), we obtain a pregallery with the same extremities that has exactly two repetitions:
\[
{C}_{0},\ldots ,{C}_{i - 1},\phi \left( {C}_{i}\right) ,\ldots ,\phi \left( {C}_{j - 1}\right) ,{C}_{j},\ldots ,{C}_{d}.
\]
So we can delete \( {C}_{i - 1} \) and \( {C}_{j} \) to obtain a gallery \( {\Gamma }^{\prime } \) of length \( d - 2 \) . The type \( {\mathbf{s}}^{\prime } \) of \( {\Gamma }^{\prime } \) is \( \left( {{s}_{1},\ldots ,{\widehat{s}}_{i},\ldots ,{\widehat{s}}_{j},\ldots ,{s}_{d}}\right) \) because \( \phi \) is type-preserving. | Yes |
Each sequence from a subset \( U \subset C\left\lbrack {a, b}\right\rbrack \) contains a uniformly convergent subsequence; i.e., \( U \) is relatively sequentially compact, if and only if it is bounded and equicontinuous | Null | No |
Theorem 13.5. We have \[{\pi }_{1}\left( {\mathrm{{GL}}\left( {n,\mathbb{C}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{U}\left( n\right) }\right) ,\;{\pi }_{1}\left( {\mathrm{{SL}}\left( {n,\mathbb{C}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{{SU}}\left( n\right) }\right) ,\] and \[{\pi }_{1}\left( {\mathrm{{SL}}\left( {n,\mathbb{R}}\right) }\right) \cong {\pi }_{1}\left( {\mathrm{{SO}}\left( n\right) }\right)\] | Proof. First, let \( G = \mathrm{{GL}}\left( {n,\mathbb{C}}\right), K = \mathrm{U}\left( n\right) \), and \( P \) be the space of positive definite Hermitian matrices. By the Cartan decomposition, multiplication \( K \times P \rightarrow G \) is a bijection, and in fact, a homeomorphism, so it will follow that \( {\pi }_{1}\left( K\right) \cong {\pi }_{1}\left( G\right) \) if we can show that \( P \) is contractible. However, the exponential map from the space \( \mathfrak{p} \) of Hermitian matrices to \( P \) is bijective (in fact, a homeomorphism) by Proposition 13.7, and the space \( \mathfrak{p} \) is a real vector space and hence contractible.
For \( G = \mathrm{{SL}}\left( {n,\mathbb{C}}\right) \), one argues similarly, with \( K = \mathrm{{SU}}\left( n\right) \) and \( P \) the space of positive definite Hermitian matrices of determinant one. The exponential map from the space \( \mathfrak{p} \) of Hermitian matrices of trace zero is again a homeomorphism of a real vector space onto \( P \) .
Finally, for \( G = \mathrm{{SL}}\left( {n,\mathbb{R}}\right) \), one takes \( K = \mathrm{{SO}}\left( n\right), P \) to be the space of positive definite real matrices of determinant one, and \( \mathfrak{p} \) to be the space of real symmetric matrices of trace zero. | Yes |
Let \( X = {\mathbb{R}}^{\mathbb{N}}, x = \left( {{x}_{0},{x}_{1},\ldots }\right) \) and \( y = \left( {{y}_{0},{y}_{1},\ldots }\right) \) . Define \( d\left( {x, y}\right) = \mathop{\sum }\limits_{n}\frac{1}{{2}^{n + 1}}\min \left\{ {\left| {{x}_{n} - {y}_{n}}\right| ,1}\right\} \). Then \( d \) is a metric on \( {\mathbb{R}}^{\mathbb{N}} \). | To see (iii), take two open balls \( B\left( {x, r}\right) \) and \( B\left( {y, s}\right) \) in \( X \). Let \( z \in \) \( B\left( {x, r}\right) \cap B\left( {y, s}\right) \). Take any \( t \) such that \( 0 < t < \min \{ r - d\left( {x, z}\right), s - d\left( {y, z}\right) \} \). By the triangle inequality we see that \( z \in B\left( {z, t}\right) \subseteq B\left( {x, r}\right) \bigcap B\left( {y, s}\right) \). It follows that the intersection of any two open balls is in \( \mathcal{T} \). It is quite easy to see now that \( \mathcal{T} \) is closed under finite intersections. | No |
Corollary 3.3.4. A point \( x \in X \) belongs to the Shilov boundary of \( A \) if and only if given any open neighbourhood \( U \) of \( x \), there exists \( f \in A \) such that \(\parallel f{\left| {}_{X \smallsetminus U}{\parallel }_{\infty } < \parallel f\right| }_{U}{\parallel }_{\infty }\) | Proof. First, let \( x \in X \smallsetminus \partial \left( A\right) \) . Then \( U = X \smallsetminus \partial \left( A\right) \) is an open neighbourhood of \( x \) and because \( \partial \left( A\right) \) is a boundary, we have for all \( f \in A \), \(\parallel f{\left| {}_{U}{\parallel }_{\infty } \leq \parallel f{\parallel }_{\infty } = \parallel f\right| }_{\partial \left( A\right) }{\parallel }_{\infty } = {\begin{Vmatrix}{\left. f\right| }_{X \smallsetminus U}\end{Vmatrix}}_{\infty }.\) Conversely, let \( x \in \partial \left( A\right) \) and suppose there exists an open neighbourhood \( U \) of \( x \) such that \(\parallel f{\left| U\right| }_{\infty } \leq \parallel f{\left| {}_{X \smallsetminus U}\right| }_{\infty }\) for all \( f \in A \) . Then \( X \smallsetminus U \) is a boundary for \( A \), so that \( \partial \left( A\right) \subseteq X \smallsetminus U \) . This contradicts \( x \in \partial \left( A\right) \) . | Yes |
Theorem 2.2.1 (The division algorithm). Let \( S = K\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) denote the polynomial ring in \( n \) variables over a field \( K \) and fix a monomial order \( < \) on \( S \) . Let \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \) be nonzero polynomials of \( S \) . Then, given a polynomial \( 0 \neq f \in S \), there exist polynomials \( {f}_{1},{f}_{2},\ldots ,{f}_{s} \) and \( {f}^{\prime } \) of \( S \) with\n\n\( f = {f}_{1}{g}_{1} + {f}_{2}{g}_{2} + \cdots + {f}_{s}{g}_{s} + {f}^{\prime }, \)(2.2)\n\nsuch that the following conditions are satisfied:\n\n(i) if \( {f}^{\prime } \neq 0 \) and if \( u \in \operatorname{supp}\left( {f}^{\prime }\right) \), then none of the initial monomials \( {\operatorname{in}}_{ < }\left( {g}_{1}\right) ,{\operatorname{in}}_{ < }\left( {g}_{2}\right) ,\ldots ,{\operatorname{in}}_{ < }\left( {g}_{s}\right) \) divides \( u \), i.e. no monomial \( u \in \operatorname{supp}\left( {f}^{\prime }\right) \) belongs to \( \left( {{\operatorname{in}}_{ < }\left( {g}_{1}\right) ,{\operatorname{in}}_{ < }\left( {g}_{2}\right) ,\ldots ,{\operatorname{in}}_{ < }\left( {g}_{s}\right) }\right) \) ;\n\n(ii) if \( {f}_{i} \neq 0 \), then\n\n\( {\operatorname{in}}_{ < }\left( f\right) \geq {\operatorname{in}}_{ < }\left( {{f}_{i}{g}_{i}}\right) \)\n\nThe right-hand side of equation (2.2) is said to be a standard expression for \( f \) with respect to \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \), and the polynomial \( {f}^{\prime } \) is said to be a remainder of \( f \) with respect to \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \) . One also says that \( f \) reduces to \( {f}^{\prime } \) with respect \( {g}_{1},\ldots ,{g}_{s} \). | Proof (of Theorem 2.2.1). Let \( I = \left( {{\operatorname{in}}_{ < }\left( {g}_{1}\right) ,\ldots ,{\operatorname{in}}_{ < }\left( {g}_{s}\right) }\right) \) . If none of the monomials \( u \in \operatorname{supp}\left( f\right) \) belongs to \( I \), then the desired expression can be obtained by setting \( {f}^{\prime } = f \) and \( {f}_{1} = \cdots = {f}_{s} = 0 \) .\n\nNow, suppose that a monomial \( u \in \operatorname{supp}\left( f\right) \) belongs to \( I \) and write \( {u}_{0} \) for the monomial which is biggest with respect to \( < \) among the monomials \( u \in \operatorname{supp}\left( f\right) \) belonging to \( I \) . Let, say, in \( {}_{ < }\left( {g}_{{i}_{0}}\right) \) divide \( {u}_{0} \) and \( {w}_{0} = {u}_{0}/{\operatorname{in}}_{ < }\left( {g}_{{i}_{0}}\right) \) . We rewrite\n\n\( f = {c}_{0}^{\prime }{c}_{{i}_{0}}^{-1}{w}_{0}{g}_{{i}_{0}} + {h}_{1} \)\n\nwhere \( {c}_{0}^{\prime } \) is the coefficient of \( {u}_{0} \) in \( f \) and \( {c}_{{i}_{0}} \) is that of \( {\operatorname{in}}_{ < }\left( {g}_{{i}_{0}}\right) \) in \( {g}_{{i}_{0}} \) . One has\n\n\( {\operatorname{in}}_{ < }\left( {{w}_{0}{g}_{{i}_{0}}}\right) = {w}_{0}{\operatorname{in}}_{ < }\left( {g}_{{i}_{0}}\right) = {u}_{0} \leq {\operatorname{in}}_{ < }\left( f\right) . \)\n\nIf either \( {h}_{1} = 0 \) or, in case of \( {h}_{1} \neq 0 \), none of the monomials \( u \in \operatorname{supp}\left( {h}_{1}\right) \) belongs to \( I \), then \( f = {c}_{0}^{\prime }{c}_{{i}_{0}}^{-1}{w}_{0}{g}_{{i}_{0}} + {h}_{1} \) is a standard expression of \( f \) with respect to \( {g}_{1},{g}_{2},\ldots ,{g}_{s} \) and \( {h}_{1} \) is a remainder of \( f \) .\n\nIf a monomial of \( \operatorname{supp}\left( {h}_{1}\right) \) belongs to \( I \) and if \( {u}_{1} \) is the monomial which is biggest with respect to \( < \) among the monomials \( u \in \operatorname{supp}\left( {h}_{1}\right) \) belonging to \( I \), then one has\n\n\( {u}_{0} > {u}_{1} \)\n\nIn fact, if a monomial \( u \) with \( u > {u}_{0}\left( { = {\operatorname{in}}_{ < }\left( {{w}_{0}{g}_{{i}_{0}}}\right) }\right) \) belongs to \( \operatorname{supp}\left( {h}_{1}\right) \) , then \( u \) must belong to \( \operatorname{supp}\left( f\right) \) . This is impossible. Moreover, \( {u}_{0} \) itself cannot belong to \( \operatorname{supp}\left( {h}_{1}\right) \) .\n\nLet, say, in \( < \left( {g}_{{i}_{1}}\right) \) divide \( {u}_{1} \) and \( {w}_{1} = {u}_{1}/{\operatorname{in}}_{ < }\left( {g}_{{i}_{1}}\right) \) . Again, we rewrite\n\n\( f = {c}_{0}^{\prime }{c}_{{i}_{0}}^{-1}{w}_{0}{g}_{i}{i}_{0} + {c}_{1}^{\prime }{c}_{{i}_{1}}^{-1}{w}_{1}{g}_{{i}_{1}} + {h}_{2}, \)\n\nwhere \( {c}_{1}^{\prime } \) is the coefficient of \( {u}_{1} \) in \( {h}_{1} \) and \( {c}_{{i}_{1}} \) is that of \( {\operatorname{in}}_{ < }\left( {g}_{{i}_{1}}\right) \) in \( {g}_{{i}_{1}} \) . One has\n\n\( {\operatorname{in}}_{ < }\left( {{w}_{1}{g}_{{i}_{1}}}\right) < {\operatorname{in}}_{ < }\left( {{w}_{0}{g}_{{i}_{0}}}\right) \leq {\operatorname{in}}_{ < }\left( f\right) . \)\n\nContinuing these procedures yields the descending sequence\n\n\( {u}_{0} > {u}_{1} > {u}_{2} > \cdots \)\n\nLemma 2.1.7 thus guarantees that these procedures will stop after a finite number of steps, say \( N \) steps, and we obtain an expression\n\n\( f = \mathop{\sum }\limits_{{q = 0}}^{{N - 1}}{c}_{q}^{\prime }{c}_{{i}_{q}}^{-1}{w}_{q}{g}_{{i}_{q}} + {h}_{N} \)\n\nwhere either \( {h}_{N} = 0 \) or, in case \( {h}_{N} \neq 0 \), none of the monomials \( u \in \operatorname{supp}\left( {h}_{N}\right) \) belongs to \( I \), and where\n\n\( {\operatorname{in}}_{ < }\left( {{w}_{q}{g}_{{i}_{q}}}\right) < \cdots < {\operatorname{in}}_{ < }\left( {{w}_{0}{g}_{{i}_{0}}}\right) \leq {\operatorname{in}}_{ < }\left( f\right) . \)\n\nThus, by letting \( \mathop{\sum }\limits_{{i = 1}}^{s}{f}_{i}{g}_{i} = \mathop{\sum }\limits_{{q = 0}}^{{N - 1}}{c}_{q}^{\prime }{c}_{{i}_{q}}^{-1}{w}_{q}{g}_{{i}_{q}} \) and \( {f}^{\prime } = {h}_{N} \), we obtain an expression \( f = \mathop{\sum }\limits_{{i = 1}}^{s}{f}_{i}{g}_{i} + {f}^{\prime } \) satisfying the conditions (i) and (ii), as desired. | Yes |
Corollary 3. For some constant \( c = c\left( f\right) \), we have \({\operatorname{ord}}_{p}\mathop{\prod }\limits_{\substack{{\text{ cond }\psi = {p}^{t}} \\ {{n}_{0} \leq t \leq n} }}B\left( {\psi ,\mu }\right) = m{p}^{n} + {\lambda n} + c\left( f\right)\) | Since \(\mathop{\prod }\limits_{\substack{{\zeta {p}^{n} = 1} \\ {\zeta \neq 1} }}\left( {\zeta - 1}\right) = {p}^{n}\), the formula is immediate, since the product taken for \( {n}_{0} \leq t \leq n \) differs by only a finite number of factors (depending on \( {n}_{0} \) ) from the product taken over all \( t \), and we can apply Corollary 2 to get the desired order. | Yes |
Proposition 5.42. Suppose \( X \) is a Hausdorff locally convex space, \( T : X \rightarrow X \) is a continuous linear transformation, and \( U \) is a barrel neighborhood of 0 subject to:\n(α) \( T\left( U\right) \) does not contain a nontrivial subspace of \( X \), and\n(β) \( T\left( U\right) \) is covered by \( N \) translates of \( \frac{1}{2}\operatorname{int}\left( U\right) \) ; that is there exists \( {w}_{1},\ldots ,{w}_{N} \in \) X for which\n\[
T\left( U\right) \subset \mathop{\bigcup }\limits_{{j = 1}}^{N}\left\lbrack {{w}_{j} + \frac{1}{2}\operatorname{int}\left( U\right) }\right\rbrack .
\]\nConsider the chain of subspaces \( {K}_{j} = \ker {\left( I - T\right) }^{j} \) :\n\[
\{ 0\} \subset {K}_{1} \subset {K}_{2} \subset \cdots
\]\nThen:\n(a) The chain stabilizes beyond \( j = N : {K}_{N} = {K}_{N + 1} = \cdots \) ;\n(b) Every \( {K}_{j} \) has dimension \( \leq N \) ; and\n(c) \( \dim {K}_{1} = \dim \ker \left( {I - T}\right) \leq \dim \left( {X/\left( {I - T}\right) \left( X\right) }\right) \) . | Proof of Proposition 5.42: This is done using a series of steps.\nStep 1: \( \dim {K}_{1} \leq N \) . Suppose \( {v}_{1},\ldots ,{v}_{n} \) is a finite, linearly independent subset of \( {K}_{1} \) . Set \( {M}_{k} = \operatorname{span}\left\{ {{v}_{1},\ldots ,{v}_{k}}\right\} \) . Since \( \left( {I - T}\right) {M}_{k} = \{ 0\} \), these spaces satisfy the hypotheses of Lemma 5.43, so \( n \leq N \) . Since \( N \) is an upper bound for any finite linearly independent subset of \( {K}_{1} \), and \( {K}_{1} \) does have a basis (which, if infinite, will have arbitrarily large finite subsets), \( {K}_{1} \) must be finite dimensional, with dimension \( \leq N \) .\nStep 2: \( \dim \left( {{K}_{j + 1}/{K}_{j}}\right) \leq \dim \left( {{K}_{j}/{K}_{j - 1}}\right) \) . Consider the composite map:\n\[
{K}_{j + 1}\overset{\left( I - T\right) }{ \rightarrow }{K}_{j}\overset{\pi }{ \rightarrow }{K}_{j}/{K}_{j - 1}
\]\nThe kernel is\n\[
\left\{ {x \in {K}_{j + 1} : \left( {I - T}\right) \left( x\right) \in {K}_{j - 1}}\right\} = \left\{ {x \in {K}_{j + 1} : {\left( I - T\right) }^{j - 1}\left( {I - T}\right) \left( x\right) = 0}\right\} = {K}_{j},
\]\nso \( \dim \left( {{K}_{j + 1}/{K}_{j}}\right) \) equals the dimension of the image of the composite, which (as a subspace) has dimension \( \leq \dim \left( {{K}_{j}/{K}_{j - 1}}\right) \) .\nStep 3: Every \( {K}_{j} \) is finite-dimensional. Induction on \( j \) . Step 1 gives the \( j = 1 \) case, while Step 2 provides the induction step.\nProof for part (a): Set \( {M}_{j} = {K}_{j} \), now known to be finite-dimensional. Once \( {K}_{j} = {K}_{j - 1} \), you get \( {K}_{j + 1} = {K}_{j} \) by Step 2, so it stabilizes beyond some \( n \), with \( {K}_{n - 1} \neq {K}_{n} = {K}_{n + 1}\cdots \) (unless all \( {K}_{j} = \{ 0\} \), in which case Proposition 5.42 is trivial). By Lemma 5.43, \( n \leq N \) .\nNow set \( K = {K}_{N} = {K}_{N + 1} = \cdots \) .\nStep 4: \( \dim \left( K\right) \leq N \) (proving part (b)). Start with a basis of \( {K}_{1} : {v}_{1},\ldots ,{v}_{l} \) . | Yes |
Example 6.4.4. Example 6.4.3(3) allows us to define a family of orientations on complex projective spaces. | We refer to the orientations obtained in this way as the standard orientations, and the ones with the opposite sign for \( \left\lbrack {\mathbb{C}{P}^{n}}\right\rbrack \) as the nonstandard orientations.
We begin with \( n = 1 \) . We have the standard generator \( {\sigma }_{1} \in {H}_{1}\left( {S}^{1}\right) \) of Remark 4.1.10.
To define an orientation of \( \mathbb{C}{P}^{1} \) it suffices to give a local orientation \( {\bar{\varphi }}_{{z}_{0}} \) at a single point \( {z}_{0} \), and we choose \( {z}_{0} \) to be the point with homogeneous coordinates \( \left\lbrack {0,1}\right\rbrack \) . We specify \( {\bar{\varphi }}_{{z}_{0}} \) by letting \( {\bar{\varphi }}_{{z}_{0}}\left( 1\right) \) be the image of \( {\sigma }_{1} \) under the sequence of isomorphisms
\[
{H}_{1}\left( {S}^{1}\right) \rightarrow {H}_{1}\left( {\mathbb{C}-\{ 0\} }\right) \rightarrow {H}_{2}\left( {\mathbb{C},\mathbb{C}-\{ 0\} }\right) \rightarrow {H}_{2}\left( {\mathbb{C}{P}^{1},\mathbb{C}{P}^{1}-\{ \left\lbrack {0,1}\right\rbrack \} }\right) .
\]
Here the first isomorphism is induced by inclusion, the second is the inverse of the boundary map in the exact sequence of the pair \( \left( {\mathbb{C},\mathbb{C}-\{ 0\} }\right) \), and the third is induced by the map \( z \mapsto \left\lbrack {z,1}\right\rbrack \) .
Given this orientation we have a fundamental class \( \left\lbrack {\mathbb{C}{P}^{1}}\right\rbrack \in {H}_{2}\left( {\mathbb{C}{P}^{1}}\right) \), and we let \( \alpha = \left\{ {\mathbb{C}{P}^{1}}\right\} \) be the fundamental cohomology class. Then for \( n > 1 \), we choose the orientation which has \( \left\{ {\mathbb{C}{P}^{n}}\right\} = {\alpha }^{n} \) as fundamental cohomology class, i.e., the orientation with fundamental class \( \left\lbrack {\mathbb{C}{P}^{1}}\right\rbrack \) specified by \( e\left( {{\alpha }^{n},\left\lbrack {\mathbb{C}{P}^{n}}\right\rbrack }\right) = 1 \). | Yes |
Prove that the integer program (8.1) can be written equivalently as \[
{z}_{I} = \max \;{cx}
\]
\[
x - y = 0
\]
\[
{A}_{1}x \leq {b}^{1}
\]
\[
{A}_{2}y \leq {b}^{2}
\]
\[
{x}_{j},{y}_{j} \in \mathbb{Z}\text{ for }j = 1,\ldots, p
\]
\[
x, y \geq 0
\] | Let \( \bar{z} \) be the optimal solution of the Lagrangian dual obtained by dualizing the constraints \( x - y = 0 \) . Prove that
\[
\bar{z} = \max \left\{ {{cx} : x \in \operatorname{conv}\left( {Q}_{1}\right) \cap \operatorname{conv}\left( {Q}_{2}\right) }\right\}
\]
where \( {Q}_{i} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{Z}}_{ + }^{p} \times {R}_{ + }^{n - p} : {A}_{i}x \leq {b}^{i}}\right\}, i = 1,2 \), assuming that \( \operatorname{conv}\left( {Q}_{1}\right) \cap \) \( \operatorname{conv}\left( {Q}_{2}\right) \) is nonempty. | No |
If \( \left( {T, S}\right) \) satisfies condition \( \left( \mathrm{C}\right) \), then \( T * S = S * T \) . | The second part of Proposition 2.6 allows us, by passing to the limit, to reduce the problem to the case of distributions with compact support, for which these properties were stated in Proposition 2.2. The reasoning is straightforward for the proof of parts 1 and 3 . We spell it out for part 2 . | No |
Proposition 5.5.6. Let \( \mathcal{B} \) be a Banach space and \( \left( {X,\mu }\right) \) a \( \sigma \) -finite measure space. (a) The set \( \left\{ {\mathop{\sum }\limits_{{j = 1}}^{m}{\chi }_{{E}_{j}}{u}_{j} : {u}_{j} \in \mathcal{B},{E}_{j} \subseteq X}\right. \) are pairwise disjoint and \( \left. {\mu \left( {E}_{j}\right) < \infty }\right\} \) is dense in \( {L}^{p}\left( {X,\mathcal{B}}\right) \) whenever \( 0 < p < \infty \) . | If \( F \in {L}^{p}\left( {X,\mathcal{B}}\right) \) for \( 0 < p \leq \infty \), then \( F \) is \( \mathcal{B} \) -measurable; thus there exists \( {X}_{0} \subseteq X \) satisfying \( \mu \left( {X \smallsetminus {X}_{0}}\right) = 0 \) and \( F\left\lbrack {X}_{0}\right\rbrack \subseteq {\mathcal{B}}_{0} \), where \( {\mathcal{B}}_{0} \) is some separable subspace of \( \mathcal{B} \) . Choose a countable dense sequence \( {\left\{ {u}_{j}\right\} }_{j = 1}^{\infty } \) of \( {\mathcal{B}}_{0} \). | No |
The vibrations of a string are modeled by the so-called wave equation \n\[
\frac{{\partial }^{2}w}{\partial {x}^{2}} = \frac{1}{{c}^{2}}\frac{{\partial }^{2}w}{\partial {t}^{2}}
\]\nwhere \( w = w\left( {x, t}\right) \) denotes the vertical elongation and \( c \) is the speed of sound in the string. | Null | No |
Show that if \( q \) is prime, then \(\frac{\varphi \left( {q - 1}\right) }{q - 1}\mathop{\sum }\limits_{{d \mid q - 1}}\frac{\mu \left( d\right) }{\varphi \left( d\right) }\mathop{\sum }\limits_{{o\left( \chi \right) = d}}\chi \left( a\right) = \left\{ \begin{array}{ll} 1 & \text{ if }a\text{ has order }q - 1 \\ 0 & \text{ otherwise,} \end{array}\right.\) | Null | No |
identify \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) and \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) . | As a result we may identify \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) and \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) . | No |
Corollary 9.25. For all \( s \in \mathbb{C}, s \notin \mathbb{N} \) , \(\Gamma \left( s\right) \Gamma \left( {1 - s}\right) = \frac{\pi }{\sin \left( {\pi s}\right) }.\) | Proof. By Theorem 9.20,\[
\Gamma \left( s\right) \Gamma \left( {-s}\right) = - \frac{1}{{s}^{2}}\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 + \frac{s}{n}\right) }^{-1}{e}^{s/n}\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - \frac{s}{n}\right) }^{-1}{e}^{-s/n}
\]
\[
= - \frac{1}{{s}^{2}}\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - \frac{{s}^{2}}{{n}^{2}}\right) }^{-1} = - \frac{\pi }{s\sin \left( {\pi s}\right) }
\]
using the classical formula
\[
\sin \left( {\pi s}\right) = {\pi s}\mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 - \frac{{s}^{2}}{{n}^{2}}}\right)
\]
(9.25)
The corollary follows because \( - {s\Gamma }\left( {-s}\right) = \Gamma \left( {1 - s}\right) \) . | Yes |
Proposition 11.2.4. Let \( \mathcal{C} \) be \( R \) -modules or Groups. Let \( Z \) be an object of \( \mathcal{C} \) and let \( {g}_{\alpha } : {X}_{\alpha } \rightarrow Z \) be a homomorphism for each \( \alpha \in \mathcal{A} \), such that \( {g}_{\alpha } = {g}_{\beta } \circ {f}_{\beta }^{\alpha } \) whenever \( \alpha \leq \beta \) . Let \( g : \lim \left\{ {{X}_{\alpha },{f}_{\alpha }^{\beta }}\right\} = : X \rightarrow Z \) be the unique homomorphism such that \( g \circ {j}_{\alpha } = {g}_{\alpha } \) for all \( \alpha \in \mathcal{A} \) . Then \( g \) is an isomorphism iff \( Z = \cup \left\{ \right. \) image \( \left. {{g}_{\alpha } \mid \alpha \in \mathcal{A}}\right\} \), and for each \( \alpha \) ker \( {g}_{\alpha } \subset \cup \left\{ \right. \) ker \( \left. {{f}_{\beta }^{\alpha } \mid \beta \geq \alpha }\right\} \) . | Null | No |
Proposition 9.2.1. Let \( \mathcal{A} \) (respectively, \( {\mathcal{A}}^{\prime } \) ) be the subalgebra of \( \operatorname{End}\left( Y\right) \) generated by \( \rho \left( K\right) \) (respectively, \( \rho \left( {K}^{\prime }\right) \) ). The following are equivalent: | The implication (2) \( \Rightarrow \) (1) follows directly from Theorem 4.2.1. Now assume that (1) holds and suppose \( {m}_{\pi ,{\pi }^{\prime }} = 1 \) for some pair \( \left( {\pi ,{\pi }^{\prime }}\right) \) . The \( {\pi }^{\prime } \) -isotypic subspace of \( Y \) (viewed as a \( {K}^{\prime } \) -module) is \( {Y}_{\left( {\pi }^{\prime }\right) } = {V}^{\pi } \otimes {V}^{{\pi }^{\prime }} \), since \( \pi \) occurs only paired with \( {\pi }^{\prime } \) in \( Y \) . Let \( T \in {\operatorname{End}}_{{K}^{\prime }}\left( Y\right) \) . Then \( T \) leaves each subspace \( {Y}_{\left( {\pi }^{\prime }\right) } \) invariant, and by the double commutant theorem (Theorem 4.1.13), \( T \) acts on \( {Y}_{\left( {\pi }^{\prime }\right) } \) as \( {T}_{\pi } \otimes {I}_{{\pi }^{\prime }} \) , where \( {T}_{\pi } \in \operatorname{End}\left( {V}^{\pi }\right) \) . Hence \( T \in \mathcal{A} \) by Corollary 4.2.4. | Yes |
Theorem 2.2. A field extension \( K \) of a field \( F \) is separable if and only if, for every field \( L \) containing \( F \), the tensor product \( K{ \otimes }_{F}L \) has no non-zero nilpotent element. | Proof. First, suppose that \( K \) is separable over \( F \), and let \( u \) be a nilpotent element of \( K{ \otimes }_{F}L \) . We shall prove that \( u = 0 \) . Clearly, there is a field \( {K}_{1} \) between \( F \) and \( K \) that is finitely field-generated over \( F \) and such that \( u \) belongs to the canonical image of \( {K}_{1}{ \otimes }_{F}L \) in \( K{ \otimes }_{F}L \), which we may identify with \( {K}_{1}{ \otimes }_{F}L \) . Therefore, we assume without loss of generality that \( K \) is finitely field-generated over \( F \) . Then, by Theorem 2.1, there is a transcendence basis \( X \) for \( K \) over \( F \) such that \( K \) is separably algebraic over \( F\left( X\right) \) . In fact, since \( K \) is also a finite algebraic extension of \( F\left( X\right) \), there is an element \( s \) in \( K \) that is separably algebraic over \( F\left( X\right) \) and such that \( K = F\left( X\right) \left\lbrack s\right\rbrack \) .
Let \( {X}^{\prime } \) be a set of independent variables over \( L \) that is in bijective correspondence with \( X \) . Then \( F\left( X\right) { \otimes }_{F}L \) may evidently be identified with an \( F\left( {X}^{\prime }\right) \) -subalgebra of the purely transcendental field extension \( L\left( {X}^{\prime }\right) \) of \( L \) . Accordingly, we may write
\[
K{ \otimes }_{F}L = K{ \otimes }_{F\left( X\right) }\left( {F\left( X\right) { \otimes }_{F}L}\right) \subset K{ \otimes }_{F\left( X\right) }L\left( {X}^{\prime }\right) ,
\]
where \( L\left( {X}^{\prime }\right) \) is viewed as an \( F\left( X\right) \) -algebra via an \( F \) -algebra isomorphism from \( F\left( X\right) \) to \( F\left( {X}^{\prime }\right) \) extending a bijection from \( X \) to \( {X}^{\prime } \) .
Let \( f \) denote the monic minimum polynomial for \( s \) relative to \( F\left( X\right) \) , but view \( f \) as a polynomial with coefficients in \( F\left( {X}^{\prime }\right) \) via the isomorphism just mentioned. Since \( s \) is separable over \( F\left( X\right) \), the polynomial \( f \) is the product of a set of mutually distinct monic irreducible factors with coefficients in \( L\left( {X}^{\prime }\right) \) . It follows from this that \( K{ \otimes }_{F\left( X\right) }L\left( {X}^{\prime }\right) \) is a direct sum of fields, one for each irreducible factor of \( f \) . Therefore, our nilpotent element \( u \) must be 0 .
Now suppose that the condition of the theorem is satisfied. In showing that \( K \) is separable over \( F \), we may assume that \( F \) is of non-zero characteristic \( p \) . We show that then the condition of Proposition 1.5 is satisfied, so that \( K \) is separable over \( F \) . Let \( \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) be an \( F \) -linearly independent subset of \( K \), and suppose that \( {c}_{1},\ldots ,{c}_{n} \) are elements of \( F \) such that
\[
\mathop{\sum }\limits_{{i = 1}}^{n}{c}_{i}{u}_{i}^{p} = 0
\]
We construct a field extension \( L = F\left\lbrack {{t}_{1},\ldots ,{t}_{n}}\right\rbrack \) of \( F \) such that \( {t}_{i}^{p} = {c}_{i} \) for each \( i \) . Then the \( p \) -th power of the element \( \mathop{\sum }\limits_{{i = 1}}^{n}{u}_{i} \otimes {t}_{i} \) of \( K{ \otimes }_{F}L \) is equal to 0, so that, because of the present assumption on \( K \), the element itself must be 0 . This gives \( {t}_{i} = 0 \), and hence \( {c}_{i} = 0 \) for each \( i \) . Our conclusion is that the set \( \left( {{u}_{1}^{p},\ldots ,{u}_{n}^{p}}\right) \) is \( F \) -linearly independent. | Yes |
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