Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Proposition 4.19. Suppose \( X \) and \( Y \) are Hausdorff topological vector spaces, and \( \left\langle {T}_{\alpha }\right\rangle \) is a net consisting of (not necessarily continuous) linear transformations from \( X \) to \( Y \) . Suppose that for all \( x \in X,\lim {T}_{\alpha }\left( x\right) \) exists: Call ... | Proof. If \( x \in X \) and \( c \) is in the base field, then\n\n\[ T\left( {cx}\right) = \lim {T}_{\alpha }\left( {cx}\right) = \lim c{T}_{\alpha }\left( x\right) = c\lim {T}_{\alpha }\left( x\right) = {cT}\left( x\right) \]\n\nsince multiplication by \( c \) is continuous. Similarly, if \( x, y \in X \), then \( {T}... | Yes |
Theorem 4.20. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( Y \) is complete. Then in the topology of bounded convergence, \( {\mathcal{L}}_{b}\left( {X, Y}\right) \), the space of bounded linear transformations from \( X \) to \( Y \), is complete. | Proof. Suppose \( \left\langle {T}_{\alpha }\right\rangle \) is a Cauchy net in \( {\mathcal{L}}_{b}\left( {X, Y}\right) \) . If \( x \in X \), then\n\n\[ \n{T}_{\alpha } - {T}_{\beta } \in {N}_{b}\left( {\{ x\}, U}\right) \Rightarrow {T}_{\alpha }\left( x\right) - {T}_{\beta }\left( x\right) \in U \n\]\n\nwhich happen... | No |
Corollary 4.21. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is bornological and \( Y \) is complete. Then \( {\mathcal{L}}_{c}\left( {X, Y}\right) \) is complete in the topology of bounded convergence. | Proof. Since \( X \) is bornological, \( {\mathcal{L}}_{c}\left( {X, Y}\right) = {\mathcal{L}}_{b}\left( {X, Y}\right) \) (Theorem 4.12) and \( {\mathcal{L}}_{b}\left( {X, Y}\right) \) is complete (Theorem 4.20). | Yes |
Corollary 4.22. The strong dual of a Hausdorff, bornological, locally convex space is complete. | Proof. The base field is complete. | No |
Theorem 4.24. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( Y \) is quasi-complete.\n\n(a) If \( X \) is infrabarreled, then \( {\mathcal{L}}_{c}\left( {X, Y}\right) \) is quasi-complete in the topology of bounded convergence.\n\n(b) If \( X \) is barreled, then \( {\mathcal{L}}_{c}\le... | Proof. Suppose \( \left\langle {T}_{\alpha }\right\rangle \) is a Cauchy net that is bounded in the topology of pointwise convergence. Then for all \( x \in X,\left\{ {{T}_{\alpha }\left( x\right) }\right\} \) is bounded by Proposition 4.14(a). Also: \( {T}_{\alpha } - {T}_{\beta } \in N\left( {\{ x\}, U}\right) \Leftr... | No |
Lemma 4.27. Suppose \( X \) is a Hausdorff locally convex space, and \( A \subset X \) . Then the following are equivalent:\n\n(i) \( A \) is precompact.\n\n(ii) For each neighborhood \( U \) of 0, there exists a finite set \( {F}_{U} \subset X \) such that \( A \subset {F}_{U} + U \) .\n\n(iii) For each neighborhood \... | Proof. (i) \( \Rightarrow \) (ii), since the definition provides \( {F}_{U} = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \subset A \) . Then (ii) \( \Rightarrow \) (iii), since finite sets are compact. Suppose (iii). Given a neighborhood \( U \) of 0, choose an open convex, balanced neighborhood \( V \) of 0 for which \... | Yes |
Corollary 4.29. Suppose \( X \) is a Hausdorff, quasi-complete, locally convex space. Then the closed convex hull of a compact set is compact. | Proof. If \( A \) is compact, then \( A \) is precompact [part (a)], hence \( \operatorname{con}\left( A\right) \) is precompact [part \( \left( \mathrm{g}\right) \) ], hence \( \operatorname{con}{\left( A\right) }^{ - } \) is precompact [part \( \left( \mathrm{d}\right) \) ] and complete [part \( \left( \mathrm{b}\rig... | No |
Corollary 4.30. Suppose \( X \) is a Hausdorff, quasi-complete, locally convex space. If \( A \) is compact in \( X \), then \( {\left( {A}^{ \circ }\right) }_{ \circ } \) is compact. | Proof. Let \( K \) denote the closed convex hull of \( A \) ; then \( K \) is compact by Corollary 4.29. Hence \( {\left( {K}^{ \circ }\right) }_{ \circ } \) is compact by Proposition 3.21. But \( A \subset K \Rightarrow {A}^{ \circ } \supset \) \( {K}^{ \circ } \Rightarrow {\left( {A}^{ \circ }\right) }_{ \circ } \sub... | Yes |
Proposition 4.32. Suppose \( X \) and \( Y \) are topological spaces, and \( f : X \rightarrow Y \) is a function. Then the following are equivalent:\n\n(i) \( f \) is nearly open.\n\n(ii) There exists a global base \( \mathcal{B} \) for the topology on \( X \) (consisting of open sets) for which \( f\left( B\right) \s... | Proof. (i) \( \Rightarrow \) (ii) Since global bases exist (the whole topology is one such!), and global bases consist of open sets \( B \) for which \( f\left( B\right) \subset \operatorname{int}\left( {f{\left( B\right) }^{ - }}\right) \) when \( f \) is nearly open.\n\n(ii) \( \Rightarrow \) (iii) Since if \( \mathc... | Yes |
Corollary 4.33. Suppose \( X \) and \( Y \) are topological vector spaces, and \( T : X \rightarrow Y \) is a linear map. Let \( {\mathcal{B}}_{0} \) denote a base for the topology of \( X \) at 0 . Then \( T \) is nearly open if, and only if, \( T{\left( B\right) }^{ - } \) is a neighborhood of 0 in \( Y \) for all \(... | Proof. If part: for all \( x \in X, x + {\mathcal{B}}_{0} \) is a base at \( x \), and for all \( B \in {\mathcal{B}}_{0} : 0 \in \) \( \operatorname{int}\left( {T{\left( B\right) }^{ - }}\right) \Rightarrow T\left( x\right) \in T\left( x\right) + \operatorname{int}\left( {T{\left( B\right) }^{ - }}\right) = \operatorn... | 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. | Proof. 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 ... | Yes |
Corollary 4.36. Suppose \( X \) is a Fréchet space, and \( Y \) is a barreled, Hausdorff, locally convex space. Suppose \( T : X \rightarrow Y \) is a continuous linear map from \( X \) onto \( Y \) . Then \( T \) induces an isomorphism of the Fréchet space \( X/\ker \left( T\right) \) with \( Y \) . | Proof. The induced map is continuous and open by Theorem 1.23(c) and (e), and is an algebraic isomorphism for the usual algebraic reasons. \( X/\ker \left( T\right) \) is Hausdorff since \( \ker T = {T}^{-1}\left( {\{ 0\} }\right) \) is closed [Theorem \( {1.23}\left( \mathrm{\;g}\right) \) ], while \( X/\ker \left( T\... | Yes |
Corollary 4.38. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is barreled and \( Y \) is a Fréchet space. Suppose \( T \) is a linear transformation from \( X \) to \( Y \), and suppose \( Z \) is a Hausdorff space and \( F : Y \rightarrow Z \) is a one-to-one continuous function ... | Proof. \( \Gamma \left( T\right) = {\left( \mathrm{{id}} \times F\right) }^{-1}\left( {\Gamma \left( {F \circ T}\right) }\right) \) is closed. | Yes |
Proposition 4.39. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is infrabarreled. Suppose \( T : X \rightarrow Y \) is a linear transformation for which \( f \circ T \in {X}^{ * } \) whenever \( f \in {Y}^{ * } \) Then \( T \) is continuous. | Proof. First of all, if \( f \in {Y}^{ * } \), then \( x \in \{ f{\} }_{ \circ } \Leftrightarrow \left| {f\left( x\right) }\right| \leq 1 \), so\n\n\[{T}^{-1}\left( {\{ f{\} }_{ \circ }}\right) = {T}^{-1}\left( {{f}^{-1}\left( {\{ z : \left| z\right| \leq 1\} }\right) }\right)\]\n\n\[= {\left( f \circ T\right) }^{-1}\l... | Yes |
Theorem 5.2. Suppose \( X \) and \( Y \) are locally convex spaces, and \( T : X \rightarrow Y \) is a continuous linear map. Letting \( {X}^{ * } \) and \( {Y}^{ * } \) denote their strong dual spaces, \( {T}^{ * } : {Y}^{ * } \rightarrow {X}^{ * } \) is a continuous linear map. Also, if \( A \subset X \), then \( T{\... | Proof. Letting \( \mathbb{F} \) denote the base field, if \( c \in \mathbb{F}, x \in X \), and \( f, g \in {Y}^{ * } \), then\n\n\[ \n{T}^{ * }\left( {cf}\right) \left( x\right) = \left( {cf}\right) \left( {T\left( x\right) }\right) = {cf}\left( {T\left( x\right) }\right) \n\]\n\n\[ \n= c\left\lbrack {{T}^{ * }\left( f... | Yes |
Proposition 5.3. Suppose \( X, Y \), and \( Z \) are locally convex spaces, and \( T \in \) \( {\mathcal{L}}_{c}\left( {X, Y}\right) \) and \( S \in {\mathcal{L}}_{c}\left( {Y, Z}\right) \) . Then \( {ST} = S \circ T \in {\mathcal{L}}_{c}\left( {X, Z}\right) \), and \( {\left( ST\right) }^{ * } = {T}^{ * }{S}^{ * } \) ... | Proof. If \( x \in X \) and \( f \in {Z}^{ * } \), then\n\n\[ \n{\left( ST\right) }^{ * }\left( f\right) \left( x\right) = f\left( {S \circ T\left( x\right) }\right) = f\left( {S\left\lbrack {T\left( x\right) }\right\rbrack }\right) \n\]\n\n\[ \n= {S}^{ * }\left( f\right) \left\lbrack {T\left( x\right) }\right\rbrack =... | Yes |
Proposition 5.4. Suppose \( X \) and \( Y \) are locally convex spaces. Equip \( {\mathcal{L}}_{c}\left( {X, Y}\right) \) and \( {\mathcal{L}}_{c}\left( {{Y}^{ * },{X}^{ * }}\right) \) with their topologies of bounded convergence. Then \( T \mapsto {T}^{ * } \) is a linear map from \( {\mathcal{L}}_{c}\left( {X, Y}\rig... | Proof. If \( S, T \in {\mathcal{L}}_{c}\left( {X, Y}\right) \), then for all \( f \in {Y}^{ * }, x \in X \), and scalar \( c \) :\n\n\[{\left( cT\right) }^{ * }\left( f\right) \left( x\right) = f\left( {{cT}\left( x\right) }\right) = {cf}\left( {T\left( x\right) }\right)\]\n\n\[= c\left\lbrack {{T}^{ * }\left( f\right)... | Yes |
Theorem 5.7. Suppose \( X \) is a locally convex space with strong dual \( {X}^{ * } \), and \( M \) is a subspace of \( X \) . Equip \( X/M \) with the quotient topology, and let \( {\left( X/M\right) }^{ * } \) denote its strong dual. Let \( \pi : X \rightarrow X/M \) denote the natural projection. Then:\n\n(a) Algeb... | Proof. (a) For all \( x \in X \) and \( f \in {\left( X/M\right) }^{ * } \), by definition \( {\pi }^{ * }\left( f\right) \left( x\right) = f\left( {\pi \left( x\right) }\right) = \) \( f\left( {x + M}\right) \) . (Again, that is all there is to it.)\n\n(b) \( {\pi }^{ * }\left( f\right) = 0 \) when \( f\left( {x + M}\... | Yes |
Lemma 5.8. Suppose \( {X}_{1},{X}_{2} \), and \( Y \) are locally convex spaces, and suppose there are continuous linear maps\n\n\[ \left\{ \begin{array}{l} {\iota }_{k} : {X}_{k} \rightarrow Y,\;k = 1,2,\text{ and } \\ {\pi }_{k} : Y \rightarrow {X}_{k}, k = 1,2 \end{array}\right. \]\n\nsubject to\n\n(i) \( {\pi }_{k}... | Proof. First of all, since \( {\pi }_{k}{\iota }_{k} \) is the identity, it is a bijection, so \( {\pi }_{k} \) is onto and \( {\iota }_{k} \) is one-to-one. But now\n\n\[ {\iota }_{1} = \left( {{\iota }_{1}{\pi }_{1} + {\iota }_{2}{\pi }_{2}}\right) {\iota }_{1} \]\n\n\[ = {\iota }_{1}\left( {{\pi }_{1}{\iota }_{1}}\r... | Yes |
Theorem 5.9. Suppose \( {X}_{1} \) and \( {X}_{2} \) are locally convex spaces. Then \( {\left( {X}_{1} \times {X}_{2}\right) }^{ * } \) is topologically isomorphic to \( {X}_{1}^{ * } \times {X}_{2}^{ * } \), where any \( \left( {f, g}\right) \in {X}_{1}^{ * } \times {X}_{2}^{ * } \) corresponds to the linear function... | Proof. We have natural maps\n\n\[ \n{\iota }_{k} : {X}_{k} \rightarrow {X}_{1} \times {X}_{2} \n\] \n\ngiven by \( {\iota }_{1}\left( {x}_{1}\right) = \left( {{x}_{1},0}\right) \) and \( {\iota }_{2}\left( {x}_{2}\right) = \left( {0,{x}_{2}}\right) \). We also have natural projections \( {\pi }_{k} : \left( {{X}_{1} \t... | Yes |
Proposition 5.12. Suppose \( X \) is a semireflexive, Hausdorff locally convex space. Then \( {X}^{ * } \) is barreled. | Proof. Suppose \( E \) is a barrel in \( {X}^{ * } \), so that \( E \) is strongly closed, convex, balanced, and absorbent. The fact that \( {J}_{X} : X \rightarrow {X}^{* * } \) is bijective says that the weak and weak-* topologies on \( {X}^{ * } \) coincide, and \( E \) is weakly closed (Theorem 3.29), hence is weak... | Yes |
Lemma 5.13. Suppose \( X \) is an infrabarreled, Hausdorff locally convex space, so that \( {J}_{X} : X \rightarrow {X}^{* * } \) has a continuous adjoint \( {J}_{X}^{ * } : {X}^{* * * } \rightarrow {X}^{ * } \) . Then \( {J}_{X}^{ * } \circ {J}_{{X}^{ * }} \) is the identity map on \( {X}^{ * } \) . | Proof. Suppose \( f \in {X}^{ * } \), and \( {J}_{{X}^{ * }}\left( f\right) = \Phi \) . If \( x \in X \), then set \( \psi = {J}_{X}\left( x\right) \) . By definition:\n\n\[ \n{J}_{X}^{ * }\left( \Phi \right) \left( x\right) = \Phi \left( {{J}_{X}\left( x\right) }\right) = \Phi \left( \psi \right) \]\n\n\[ = {J}_{{X}^{... | Yes |
Proposition 5.14. Suppose \( X \) is a reflexive Hausdorff locally convex space. Then \( {X}^{ * } \) is also reflexive. | Proof. By assumption \( {J}_{X} : X \rightarrow {X}^{* * } \) is a topological isomorphism with inverse \( {J}_{X}^{-1} : {X}^{* * } \rightarrow X \), so \( {J}_{X}^{ * } : {X}^{* * * } \rightarrow {X}^{ * } \) is a topological isomorphism with inverse \( {\left( {J}_{X}^{-1}\right) }^{ * } : {X}^{ * } \rightarrow {X}^... | Yes |
Corollary 5.15. Suppose \( X \) is a semireflexive Hausdorff locally convex space. Then the following are equivalent:\n\n(i) \( X \) is reflexive.\n\n(ii) \( X \) is infrabarreled.\n\n(iii) \( X \) is barreled. | Proof. (iii) \( \Rightarrow \) (ii) trivially, while (ii) \( \Rightarrow \) (i) by Theorem 5.10. Suppose (i): that is, suppose \( X \) is reflexive. Then \( {X}^{ * } \) is reflexive by Proposition 5.14, so \( {X}^{* * } \) is barreled by Proposition 5.12. Hence \( X \) is barreled since \( X \) is topologically isomor... | No |
Proposition 5.16. Suppose \( X \) is a Hausdorff locally convex space. Then \( X \) is semireflexive if, and only if, bounded, closed, convex subsets of \( X \) are weakly compact. | Proof. First, suppose \( X \) is semireflexive, and \( B \) is bounded, closed, and convex in \( X \) . Then \( B \subset {\left( {B}^{ \circ }\right) }_{ \circ } \), and \( {\left( {B}^{ \circ }\right) }_{ \circ } \) is weakly closed, convex, bounded, and balanced. Set \( E = {B}^{ \circ } \) ; then \( E \) is a stron... | Yes |
Corollary 5.17. Suppose \( X \) is a Hausdorff locally convex space, and suppose \( {X}^{ * } \) is semireflexive. Then \( X \) is infrabarreled if, and only if, \( X \) is a Mackey space. | Proof. Infrabarreled spaces are Mackey spaces by Corollary 4.9, so suppose \( X \) is a Mackey space and \( {X}^{ * } \) is semireflexive. If \( D \) is strongly bounded in \( {X}^{ * } \), then \( E = \) \( {\left( {D}^{ \circ }\right) }_{ \circ } \) is also strongly bounded, as well as weakly closed, convex, and bala... | Yes |
Proposition 5.18. Suppose \( X \) is a Hausdorff locally convex space and suppose \( X \) is a quasi-complete Mackey space. Finally, suppose \( {X}^{ * } \) is semireflexive. Then \( X \) and \( {X}^{ * } \) are both reflexive. | Proof. Since \( {X}^{ * } \) is semireflexive and \( X \) is a Mackey space, \( {J}_{X} : X \rightarrow {J}_{X}\left( X\right) \) is a topological isomorphism by Corollary 5.17 and Theorem 5.10(d). Hence, in view of Proposition 5.14 ( \( X \) is reflexive \( \Rightarrow {X}^{ * } \) is reflexive), it suffices to show t... | Yes |
Theorem 5.20 (Montel’s Theorem, from Complex Analysis). If \( U \) is a region in \( \mathbb{C} \), then the Fréchet space \( \mathcal{H}\left( U\right) \), consisting of holomorphic functions on \( U \), is a Montel space. | Proof. Suppose \( C \) is a closed, bounded subset of \( \mathcal{H}\left( U\right) \) . Since \( \mathcal{H}\left( U\right) \) is metrizable, it suffices to show that any sequence \( \left\langle {f}_{n}\right\rangle \) from \( C \) has a subsequence that converges to a function in \( C \) . But all seminorms are boun... | No |
Proposition 5.21. Montel spaces are reflexive. | Proof. Suppose \( X \) is a Montel space, and suppose \( C \) is a weakly bounded, closed, convex subset of \( X \) . Then \( C \) is originally bounded (Corollary 3.31), and originally closed (the original topology is stronger), hence is originally compact since \( X \) is a Montel space. That means that \( C \) is we... | Yes |
Proposition 5.22. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( \left\langle {{T}_{\alpha } : \alpha \in D}\right\rangle \) is a net in \( {\mathcal{L}}_{c}\left( {X, Y}\right) \) that has the following properties:\n\n(a) There is a \( T \in {\mathcal{L}}_{c}\left( {X, Y}\right) \) for... | Proof. Suppose \( K \) is compact in \( X \) and \( U \) is a barrel neighborhood of 0 in \( Y \) . Set\n\n\[ V = \left( {\mathop{\bigcap }\limits_{{\alpha \in D}}{T}_{\alpha }^{-1}\left( {\frac{1}{3}U}\right) }\right) \bigcap {T}^{-1}\left( {\frac{1}{3}U}\right) \]\n\nThen \( V \) is a barrel neighborhood of 0 in \( X... | Yes |
Theorem 5.23. The strong dual of a Montel space is another Montel space. | Proof. Suppose \( X \) is a Montel space. Then \( {X}^{ * } \) is barreled since \( X \) is reflexive (Propositions 5.12 and 5.21). It remains to show that closed, strongly bounded subsets of \( {X}^{ * } \) are strongly compact.\n\nSuppose \( C \) is a strongly closed, bounded subset of \( {X}^{ * } \) . Then \( C \) ... | Yes |
Proposition 5.24. Suppose \( X \) is a Montel space, and \( B \) is a bounded subset of \( X \) . Then the original topology and the weak topology coincide on \( B \) . | Proof. It suffices to show that\n\n\[ \left( {{B}^{ - }\text{, original topology}}\right) \rightarrow \left( {{B}^{ - }\text{, weak topology}}\right) \]\n\nis a homeomorphism. But this arrow is continuous; the space on the left is compact; and the space on the right is Hausdorff. Hence this arrow is a homeomorphism by ... | No |
Theorem 5.25 (Krein-Milman). Suppose \( X \) is a Hausdorff locally convex space over \( \mathbb{R} \), and \( K \) is a compact convex subset of \( X \) . Then \( K \) is the closed convex hull of its set of extreme points. | Proof. Let \( L \) denote the closed convex hull of the set of extreme points of \( K \) . Then \( L \) is closed and convex, and \( L \subset K \), so \( L \) is compact. Suppose \( p \in K - L \) . Then there exists \( f \in {X}^{ * } \) and \( {r}_{0} \in \mathbb{R} \) for which \( f\left( x\right) < {r}_{0} \) when... | Yes |
Proposition 5.26 (Milman). Suppose \( X \) is a Hausdorff locally convex space, and \( K \) is a compact subset of \( X \) with a closed convex hull \( \operatorname{con}{\left( K\right) }^{ - } \) that is also compact. Then all extreme points of \( \operatorname{con}{\left( K\right) }^{ - } \) belong to \( K \) . | Proof. Suppose not; suppose \( p \) is an extreme point of \( \operatorname{con}{\left( K\right) }^{ - } \) that is not in \( K \) . Let \( U \) be a barrel neighborhood of 0 for which \( \left( {p + U}\right) \cap K = \varnothing \) . The set of all \( x + \operatorname{int}\left( U\right) \), with \( x \in K \), cove... | Yes |
Proposition 5.29. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and \( T \) : \( X \rightarrow Y \) is a linear transformation. Letting \( \Gamma \left( ?\right) \) denote the graph, and identifying \( {\left( X \times Y\right) }^{ * } \) with \( {Y}^{ * } \times {X}^{ * } \) via Theorem 5.9 (note th... | Proof. Observe that \( \left( {f, g}\right) \in \Gamma {\left( T\right) }^{ \bot } \) if, and only if, \( f\left( x\right) + g\left( {T\left( x\right) }\right) = 0 \) for all \( x \in X \) . That is, if and only if \( g\left( {T\left( x\right) }\right) = - f\left( x\right) \) . Since \( f \) is continuous, automaticall... | Yes |
Corollary 5.34 (Open Mapping Theorem of Ptak). Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( Y \) is barreled and \( X \) is B-complete. If \( T : X \rightarrow Y \) is continuous and onto, then \( T \) is an open map. | Proof. Replace \( X \) with \( X/\ker \left( T\right) \), a space that is also \( B \) -complete, hence is \( {B}_{r} \) -complete, by Proposition 5.32. We have a continuous algebraic isomorphism\n\n\[ X/\ker \left( T\right) \overset{{T}_{0}}{ \rightarrow }Y \]\n\nby Theorem 1.23(c); to show that \( T \) is an open map... | Yes |
Theorem 5.36. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and \( T \) : \( X \rightarrow Y \) is a continuous linear transformation. Then\n\n(a) \( T{\left( X\right) }^{ \bot } = \ker {T}^{ * } \) ,\n\n(b) \( {\left( \ker {T}^{ * }\right) }_{ \bot } = T{\left( X\right) }^{ - } \) ,\n\n(c) \( {T}^{ ... | Proof. In the third sentence of Theorem 5.2, setting \( A = X \) gives \( T{\left( X\right) }^{ \circ } = \) \( {\left( {T}^{ * }\right) }^{-1}\left( {X}^{ \circ }\right) = \ker {T}^{ * } \) ; but \( T{\left( X\right) }^{ \circ } = T{\left( X\right) }^{ \bot } \) since \( T\left( X\right) \) is a subspace. This gives (... | Yes |
Proposition 5.37. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is B-complete (or a Fréchet space) and \( Y \) has the property that every closed subspace is barreled. If \( T : X \rightarrow Y \) is a continuous linear transformation with closed range, then \( {T}^{ * } \) has we... | Proof. Assuming \( T\left( X\right) \) is closed forces \( T\left( X\right) \) to be barreled, so we now have that \( T \) is an open map (Corollary 5.34 or Theorem 4.35) onto \( T\left( X\right) \) . Hence the induced map \( {T}_{0} : X/\ker \left( T\right) \rightarrow T\left( X\right) \) is a topological isomorphism ... | Yes |
Proposition 5.38. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is barreled and \( B \) -complete (or is a Fréchet space) and \( Y \) is first countable. If \( T : X \rightarrow Y \) is a continuous linear transformation, and if \( {T}^{ * } \) has weak-* closed range, then \( T \... | Proof. (Very weird): Assuming that \( {T}^{ * } \) has weak-* closed range, we have that \( {T}^{ * }\left( {Y}^{ * }\right) = {\left( \ker T\right) }^{ \bot } \) (Theorem 5.36(d)), so that \( {T}^{ * }\left( {Y}^{ * }\right) \approx {\left( X/\ker \left( T\right) \right) }^{ * } \) by Theorem 5.7. Let \( {\tau }_{1} \... | Yes |
Proposition 5.39. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces, and suppose \( X \) is a Fréchet space and \( Y \) is infrabarreled. Suppose \( T : X \rightarrow Y \) is a continuous linear map for which \( T{\left( X\right) }^{ - } = Y \) . Then \( {T}^{ * } \) is one-to-one. If \( {\left( {T}^{ * }... | Proof. \( {T}^{ * } \) is one-to-one since \( T \) has dense range (Theorem 5.36(a)).\n\nSuppose \( U \) is a barrel neighborhood of 0 in \( X \) . Then \( {U}^{ \circ } \) is equicontinuous, hence is strongly bounded in \( {X}^{ * } \) [Theorem 4.16(a)], so \( {U}^{ \circ }\bigcap {T}^{ * }\left( {Y}^{ * }\right) \) i... | Yes |
Proposition 5.40. Suppose \( X \) and \( Y \) are Hausdorff locally convex spaces over \( \mathbb{R} \) , and \( T : X \rightarrow Y \) is a compact linear map. Then \( T \) is continuous. If \( U \) is a barrel neighborhood of 0 in \( X \) for which \( T{\left( U\right) }^{ - } \) is compact, and \( V \) is a neighbor... | Proof. Suppose \( T \) is compact, and \( U \) is a neighborhood of 0 in \( X \) for which \( T{\left( U\right) }^{ - } \) is compact. If \( V \) is a neighborhood of 0 in \( Y \), then there exists \( c > 0 \) such that \( T{\left( U\right) }^{ - } \subset {cV} \) since \( T{\left( U\right) }^{ - } \) is bounded, so \... | No |
Lemma 5.43. Assume \( X, T \), and \( U \) are as in Proposition 5.42. Suppose\n\n\[ 0 = {M}_{0} \subsetneqq {M}_{1} \subsetneqq {M}_{2} \subsetneqq \cdots \subsetneqq {M}_{n} \]\n\nis a chain of finite dimensional subspaces of \( X \) for which \( \left( {I - T}\right) {M}_{k} \subset {M}_{k - 1} \) when \( k \geq 1 \... | Proof of Lemma 5.43. The first thing to note is that \( {p}_{U} \) is a norm on each \( {M}_{k} \) . This is by induction on \( k \), and is trivial when \( k = 0 \) . As for \( k \rightarrow k + 1 \), suppose \( x \in {M}_{k + 1} \) and \( {p}_{U}\left( x\right) = 0 \) . Letting \( \mathbb{F} \) denote the scalar fiel... | Yes |
Theorem 6.1. Suppose \( X \) and \( Y \) are Fréchet spaces, \( T \in {\mathcal{L}}_{c}\left( {X, Y}\right) \), and \( {T}^{ * } \) has strongly closed range. Then \( T \) has closed range. | Proof. First, replace \( Y \) with \( T{\left( X\right) }^{ - } \), which is also a Fréchet space. We have (dually):\n\n\[ \left\lbrack \begin{array}{l} X\overset{T}{ \rightarrow }T{\left( X\right) }^{ - } \hookrightarrow Y \\ {X}^{ * }\overset{{\text{ new }}^{{T}^{ * }}}{ \leftarrow }{Y}^{ * }/T{\left( X\right) }^{ \b... | Yes |
Corollary 6.2. Suppose \( X \) and \( Y \) are Fréchet spaces, and \( T \in {\mathcal{L}}_{c}\left( {X, Y}\right) \) . The following are equivalent:\n\n(i) \( T \) has closed range.\n\n(ii) \( {T}^{ * } \) has weak-* closed range.\n\n(iii) \( {T}^{ * } \) has strongly closed range. | Proof. (iii) \( \Rightarrow \) (i) is Theorem 6.1. (i) \( \Rightarrow \) (ii) is Proposition 5.37. (ii) \( \Rightarrow \) (iii) is trivial. | Yes |
Lemma 6.4. Suppose \( X \) is a locally convex space. Let \( {\tau }^{ * } \) denote the weak-* topology on \( {X}^{ * } \) . Then:\n\n(a) \( C \) is almost weak-* open if, and only if, \( C \cap E \) is relatively weak-* open in \( E \) for all equicontinuous sets \( E \) . | Proof. (a) If \( C \) is almost weak-* open, and \( E \) is equicontinuous, then there exists a barrel neighborhood \( U \) of 0 such that \( {U}^{ \circ } \supset E \) . Hence \( C \cap E = \) \( \left( {C\bigcap {U}^{ \circ }}\right) \bigcap E \) is relatively weak-* open in \( E \) since \( C\bigcap {U}^{ \circ } \)... | Yes |
Lemma 6.5. Suppose \( X \) is a locally convex space, and \( {W}^{ * } \) is an almost weak-* open neighborhood of 0 in \( {X}^{ * } \) . Suppose \( U \) and \( V \) are neighborhoods of 0 in \( X \) for which \( U \subset V \) and \( {V}^{ \circ } \subset {W}^{ * } \) . Then there exists a finite set \( F \subset V \)... | Proof. \( {U}^{ \circ } \) is equicontinuous, as are all \( {\left( U \cup F\right) }^{ \circ } \), so \( {U}^{ \circ } - {W}^{ * } \) and all \( {\left( U \cup F\right) }^{ \circ } - {W}^{ * } \) are relatively weak-* closed in the weak-* compact (Banach-Alaoglu) set \( {U}^{ \circ } \) . So suppose not; suppose all \... | Yes |
Theorem 6.6 (Banach-Dieudonné). Suppose \( X \) is a first countable Hausdorff locally convex space. Then \( {\tau }_{a} = {\tau }_{N} = {\tau }_{K} \) . | Proof. We already know that \( {\tau }_{N} \subset {\tau }_{K} \subset {\tau }_{a} \) (Lemma 6.4), and all are translation invariant, so it suffices to show that if \( {W}^{ * } \) is an open \( {\tau }_{a} \) -neighborhood of 0, then \( {W}^{ * } \) contains a \( {\tau }_{N} \) -neighborhood of 0 .\n\nLet \( {U}_{1} \... | Yes |
Corollary 6.7. Suppose \( X \) is a Fréchet space. Then any almost weak-* continuous linear functional on \( {X}^{ * } \) is evaluation at a point of \( X \) . | Proof. Since \( X \) is complete, the closed convex hull of a compact set is compact (Theorem 4.28), so (taking convex hulls) \( {\tau }_{K} \) can be defined using the polars of all compact convex sets. These sets are weakly compact and convex, and the (finer) topology defined by the polars of weakly compact convex se... | No |
Corollary 6.8 (Krein-Smulian I). Suppose \( X \) is a Fréchet space, and \( C \) is a convex subset of \( {X}^{ * } \) . Then \( C \) is weak-* closed in \( {X}^{ * } \) if, and only if, \( C \cap {U}^{ \circ } \) is weak-* closed for every barrel neighborhood \( U \) of 0 in \( X \) . | Proof. This is just Theorem 3.29 applied to \( \left( {{X}^{ * },{\tau }_{a}}\right) \) . | No |
Proposition 6.9. Suppose \( X \) is a Hausdorff locally convex space, and suppose \( D \) is a weak-* closed, convex, balanced, absorbent subset of \( {X}^{ * } \) . Then \( D \) is a strong neighborhood of 0 in \( {X}^{ * } \) . | Proof. Since \( D \) is weak-* closed, convex, balanced, and nonempty, \( D = {\left( {D}_{ \circ }\right) }^{ \circ } \) . Set \( A = {D}_{ \circ } \) ; it suffices to show that \( A \) is bounded. But \( {A}^{ \circ } = D \) is absorbent, so \( A \) is bounded by Corollary 3.31. | Yes |
Proposition 6.10. Suppose \( X \) is a Hausdorff, first countable, locally convex space, and suppose \( {V}_{n} \) is a sequence of convex, balanced, strong neighborhoods of 0 in \( {X}^{ * } \) . Then: If \( \bigcap {V}_{n} \) absorbs all strongly bounded sets, then \( \bigcap {V}_{n} \) is a strong neighborhood of 0 ... | Proof. Set \( V = \bigcap {V}_{n} \) . Let \( {U}_{1} \supset {U}_{2} \supset \cdots \) be a base for the topology of \( X \) at 0 . Then each \( {U}_{n}^{ \circ } \) is equicontinuous, hence is strongly bounded. Choose \( {t}_{n} > 0 \) so that \( {t}_{n}{U}_{n}^{ \circ } \subset \frac{1}{2}V \), and choose a bounded ... | No |
Lemma 6.11. Suppose \( X \) is a Hausdorff, first countable, locally convex space. Suppose:\n\n( \( \alpha ){U}_{1} \supset {U}_{2} \supset \cdots \) is a base for the topology of \( X \) at 0,\n\n(β) \( {t}_{1},{t}_{2},\ldots \) is a sequence of positive real numbers,\n\n( \( \gamma ){A}_{1},{A}_{2},\ldots \) is a seq... | Proof. Each \( {t}_{j}{U}_{j}^{ \circ } \) is weak-* compact (Banach-Alaoglu), so \( \operatorname{con}\left( {{t}_{1}{U}_{1}^{ \circ } \cup \cdots \cup {t}_{n}{U}_{n}^{ \circ }}\right) \) is weak-* compact by induction on \( n \) . (Usual business: The convex hull of \( C \cup D \) is compact when \( C \) and \( D \) ... | Yes |
Theorem 6.12. The second dual of a Fréchet space is another Fréchet space. | Proof. Let \( {U}_{1} \supset {U}_{2} \supset \cdots \) be a neighborhood base for the topology of \( X \) at 0 . Then each \( {U}_{n}^{ \circ } \) is equicontinuous, hence strongly bounded in \( {X}^{ * } \) (Theorem 4.16(a)). Thus each \( {U}_{n}^{\circ \circ } \) is a strong neighborhood of 0 in \( {X}^{* * } \) . I... | Yes |
Theorem 6.13. Suppose \( X \) is a Fréchet space. Then the following are equivalent:\n\n(i) \( {X}^{ * } \) is barreled.\n\n(ii) \( {X}^{ * } \) is infrabarreled.\n\niii) \( {X}^{ * } \) is bornological. | Proof. As always,(i) \( \Rightarrow \) (ii) \( \Leftarrow \) (iii). Also,(ii) \( \Rightarrow \) (i) since \( {X}^{ * } \) is complete, by Corollaries 4.22 and 4.8. It remains to prove that (ii) \( \Rightarrow \) (iii) for duals of Fréchet spaces. Suppose \( {X}^{ * } \) is infrabarreled, and suppose \( C \) is a convex... | Yes |
Lemma 1.2.2. Homotopy is an equivalence relation. | Proof. Reflexive: \( \;f \) is homotopic to \( f \) via the homotopy of waiting (i.e., changing nothing) for one unit of time.\n\nSymmetric: If \( f \) is homotopic to \( g \), then \( g \) is homotopic to \( f \) via the homotopy of running the original homotopy backwards in time.\n\nTransitive: If \( f \) is homotopi... | Yes |
Let us regard \( X : {\mathbb{R}}^{n} - \{ \left( {0,\ldots ,0}\right) \} \) as the space of nonzero vectors \( \left\{ {v \in {\mathbb{R}}^{n} \mid v \neq 0}\right\} \) . Then \( A = {S}^{n - 1} = \left\{ {v \in {\mathbb{R}}^{n} \mid \parallel v\parallel = 1}\right\} \) is a subspace of \( X \), and is a strong deform... | The map\n\n\[ F : X \times I \rightarrow A \]\n\ngiven by\n\n\[ F\left( {v, t}\right) = \parallel v{\parallel }^{t}\left( {v/\parallel v\parallel }\right) \]\ngives a homotopy rel \( A \) from the retraction \( {f}_{0}\left( v\right) = v/\parallel v\parallel \) to the identity map \( {f}_{1}\left( v\right) = v \) \( \d... | Yes |
Lemma 1.2.12. For any space \( X,{cX} \) is contractible. | Proof. Let \( F : {cX} \times I \rightarrow {cX} \) be defined by\n\n\[ F\left( {\left( {x, s}\right), t}\right) = \left( {x,\max \left( {s, t}\right) }\right) . \]\n | Yes |
Lemma 2.1.2. The fundamental group \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is a group. | Proof. The identity element of this group is represented by the constant map \( i : {S}^{1} \rightarrow \) \( \left\{ {x}_{0}\right\} \) and the inverse of \( f : \left( {{S}^{1},1}\right) \rightarrow \left( {X,{x}_{0}}\right) \) is represented by the map \( g : \left( {{S}^{1},1}\right) \rightarrow \) \( \left( {X,{x}... | Yes |
Lemma 2.1.3. Let \( {x}_{0},{x}_{1} \in X \) . Choose a path \( \varphi \) from \( {x}_{0} \) to \( {x}_{1} \), i.e., a map \( \varphi : I \rightarrow X \) with \( \varphi \left( 0\right) = {x}_{0} \) and \( \varphi \left( 1\right) = {x}_{1} \) . Let \( \bar{\varphi } : I \rightarrow X \) be the map given by \( \bar{\v... | \[ g\left( t\right) = \left\{ \begin{array}{ll} \varphi \left( {3t}\right) & 0 \leq t \leq \frac{1}{3} \\ f\left( {{3t} - 1}\right) & \frac{1}{3} \leq t \leq \frac{2}{3} \\ \bar{\varphi }\left( {{3t} - 2}\right) & \frac{2}{3} \leq t \leq 1 \end{array}\right. \] | Yes |
Theorem 2.1.5. Let \( f : X \rightarrow Y \) with \( f\left( {x}_{0}\right) = {y}_{0} \). If \( f \) is a homotopy equivalence, then \( {f}_{ * } : {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {Y,{y}_{0}}\right) \) is an isomorphism. | Proof. Suppose that \( g : Y \rightarrow X \) with \( g\left( {y}_{0}\right) = {x}_{0} \), that \( {gf} : X \rightarrow X \) is homotopic to the identity rel \( {x}_{0} \), and that \( {fg} : Y \rightarrow Y \) is homotopic to the identity rel \( {y}_{0} \). Then the theorem is very easy to prove. But we are not making... | No |
Corollary 2.1.6. Let \( X \) be a contractible space. Then \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is the trivial group. | Proof. This is clearly true if \( X \) is the space consisting of the point \( {x}_{0} \) alone, as then every \( f : \left( {{S}^{1},1}\right) \rightarrow \left( {X,{x}_{0}}\right) \) is the constant map to the point \( {x}_{0} \) . Then it is also true for \( X \) contractible by Theorem 2.1.5. | No |
Let \( G = \mathbb{Z} \) act on \( Y = \mathbb{R} \) by \( n\left( r\right) = r + n, n \in G \) and \( r \in \mathbb{R} \) . Let \( X = G \smallsetminus Y \) . Then \( X \) is homeomorphic to \( {S}^{1} \), so we see that \( {\pi }_{1}\left( {{S}^{1},1}\right) = \mathbb{Z} \). | Note that we may identify the covering projection \( p : Y \rightarrow X \) with the covering projection in Example 2.2.3(iia). It is worth being completely explicit here. Let \( \pi : \mathbb{R} \rightarrow {S}^{1} \) by \( \pi \left( t\right) = \exp \left( {2\pi it}\right) \) . Let \( d \) be an integer and let \( {\... | Yes |
Theorem 2.2.8. Let \( p : Y \rightarrow X \) be a covering projection and let \( {y}_{0} \in Y \) and \( {x}_{0} \in X \) be points with \( p\left( {y}_{0}\right) = {x}_{0} \). Let \( E \) be an arbitrary connected and locally path connected space and let \( {e}_{0} \) be a point in \( E \). Let \( f : \left( {E,{e}_{0... | (Since \( f = p\widetilde{f} \), the condition in the theorem is obviously necessary. The point of the theorem is that it is sufficient.) | Yes |
Corollary 2.2.10. Let \( {Y}_{1} \) and \( {Y}_{2} \) be path connected and let \( {p}_{1} : \left( {{Y}_{1},{y}_{0}^{1}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) and \( {p}_{2} : \left( {{Y}_{2},{y}_{0}^{2}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) be covering projections. Then \( \left( {{Y}_{1},{y}_{0}... | \[ {\left( {p}_{1}\right) }_{ * }\left( {{\pi }_{1}\left( {{Y}_{1},{y}_{0}^{1}}\right) }\right) = {\left( {p}_{2}\right) }_{ * }\left( {{\pi }_{1}\left( {{Y}_{2},{y}_{0}^{2}}\right) }\right) . \] | Yes |
Corollary 2.2.21. Under Hypotheses 2.2.17:\n\nEvery \( X \) has a simply-connected cover \( p : \widetilde{X} \rightarrow X \), unique up to equivalence. \( \widetilde{X} \) is the universal cover of \( X \), and \( X \) is the quotient of \( \widetilde{X} \) by the group of covering translations. Also, if \( Y \) is a... | Proof. This is a direct consequence of Theorem 2.2.19, and our earlier results, taking \( H \) to be the trivial subgroup of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) . | Yes |
Theorem 2.3.1. Let \( X = {X}_{1} \cup {X}_{2} \) and suppose that \( {X}_{1},{X}_{2} \), and \( A = {X}_{1} \cap {X}_{2} \) are all open, path connected subsets of \( X \) . Let \( {x}_{0} \in A \) . Then \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is the free product with amalgamation | \[ {\pi }_{1}\left( {X,{x}_{0}}\right) = {\pi }_{1}\left( {{X}_{1},{x}_{0}}\right) { * }_{{\pi }_{1}\left( {A,{x}_{0}}\right) }{\pi }_{1}\left( {{X}_{2},{x}_{0}}\right) . \] In other words, if \( {i}_{1} : A \rightarrow {X}_{1} \) and \( {i}_{2} : A \rightarrow {X}_{2} \) are the inclusions, then \( {\pi }_{1}\left( {X... | Yes |
Corollary 2.3.3. For \( n > 1 \), the \( n \) -sphere \( {S}^{n} \) is simply connected. | Proof. We regard \( {S}^{n} \) as the unit sphere in \( {\mathbb{R}}^{n + 1} \) . Let \( {X}_{1} = {S}^{n} - \{ \left( {0,0,\ldots ,0,1}\right) \} \) and \( {X}_{2} = {S}^{n} - \{ \left( {0,0,\ldots ,0, - 1}\right) \} \) . Then \( {X}_{1} \) and \( {X}_{2} \) are both homeomorphic to \( {\mathring{D}}^{n} \), so are pa... | Yes |
Corollary 2.3.6. The fundamental group \( {\pi }_{1}\left( {{R}_{n},{r}_{0}}\right) \) is the free group on the \( n \) elements \( {\alpha }_{k} = {\left( {i}_{k}\right) }_{ * }\left( {g}_{k}\right) \), where \( {g}_{k} \) is a generator of \( {\pi }_{1}\left( {{\left( {S}^{1}\right) }_{k},{\left( 1\right) }_{k}}\righ... | Proof. We proceed by induction on \( n \) .\n\nFor \( n = 1 \) this is Example 2.2.7.\n\nNow suppose that \( n \geq 1 \) and that the theorem is true for \( n \) . Write \( {R}_{n + 1} = {X}_{1} \cup {X}_{2} \) where:\n\n\[ \n{X}_{1} = \mathop{\bigcup }\limits_{{k = 1}}^{n}{\left( {S}^{1}\right) }_{k} \cup \left\{ {{\l... | Yes |
Theorem 2.4.4. Let \( C \) be a connected 1-complex, \( v \) a vertex of \( C \), and \( T \) a maximal tree in \( C \) . Then \( {\pi }_{1}\left( {C, v}\right) \) is a free group with generators in 1-1 correspondence with the edges of \( C \) not in \( T \) . A generator is obtained from such an edge \( E \) as follow... | Note that this theorem generalizes Corollary 2.3.6 in the case of a finite 1- complex, and can be proved in a very similar fashion. It remains true for infinite 1-complexes as well. | No |
Corollary 2.4.5. Let \( H \) be a subgroup of a free group \( G \) . Then \( H \) is a free group. | Proof. Consider a rose \( R \) whose edges are in 1-1 correspondence with the generators of \( G \) . By Theorem 2.2.19, there is a cover \( \widetilde{R} \) with \( {\pi }_{1}\left( {\widetilde{R},\widetilde{v}}\right) = H \) . But it is easy to see that, since a covering projection \( p : \widetilde{R} \rightarrow R ... | Yes |
Corollary 2.4.6. Let \( G \) be a free group on \( k \) elements and let \( H \) be a subgroup of \( G \) of index \( n \) . Then \( H \) is a free group on \( \left( {k - 1}\right) n + 1 \) elements. | Proof. We may consider \( G \) to be the fundamental group \( {\pi }_{1}\left( {R, v}\right) \), where \( R \) is a \( k \) - leafed rose. Then \( H \) is the fundamental group of an \( n \) -fold cover \( {\pi }_{1}\left( {\widetilde{R},\widetilde{v}}\right) \) . Now \( R \) has 1 vertex and \( k \) edges, so \( \wide... | Yes |
Theorem 2.5.1. Let \( X \) be a path connected space. Then there is a 1-1 correspondence between conjugacy classes of elements of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) and \( \pi \left( X\right) = \) \{homotopy classes of maps \( {S}^{1} \rightarrow X \) \}. | Proof. Let \( \Phi : {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow \pi \left( X\right) \) be the map given by \ | No |
Let \( X \) be the subspace of \( {\mathbb{R}}^{2} \) consisting of the closed line segments joining the points \( \left( {1/n,0}\right) \) to \( \left( {0,1}\right) \) for each positive integer \( n \), and also the closed line segment joining \( \left( {0,0}\right) \) to \( \left( {0,1}\right) \) . Give \( X \) the t... | Let \( A \) be the subspace of \( X \) consisting of the closed line segment joining \( \left( {0,0}\right) \) to \( \left( {0,1}\right) \) . Then \( X \) and \( A \) are both contractible to the point \( \left( {0,1}\right) \) by a homotopy leaving that point fixed. It then follows that \( A \) is a deformation retrac... | No |
Theorem 3.2.2. Let \( f : \\left( {X, A}\\right) \\rightarrow \\left( {Y, B}\\right) \) be a map of pairs and suppose that both \( f \) : \( X \\rightarrow Y \) and \( f \\mid A : A \\rightarrow B \) are homotopy equivalences. Then \( {f}_{i} : {H}_{i}\\left( {X, A}\\right) \\rightarrow {H}_{i}\\left( {Y, B}\\right) \)... | Proof. We have the commutative diagram of exact sequences:\n\n\n\nThe first, second, fourth, and fifth vertical arrows are isomorphisms. Hence, by Lemma A.1.8, so is the third. | Yes |
Theorem 3.2.4. Let \( X \) be a nonempty space and let \( {x}_{0} \) be an arbitrary point of \( X \) . Then for each \( i \) ,\n\n(i) \( {\widetilde{H}}_{i}\left( X\right) \cong {H}_{i}\left( {X,{x}_{0}}\right) \),\n\n(ii) \( {H}_{i}\left( X\right) \cong {H}_{i}\left( {x}_{0}\right) \oplus {\widetilde{H}}_{i}\left( X\... | Proof. As \( {x}_{0} \) is a retract of \( X \), this is a special case of Lemma 3.2.1. | No |
Lemma 3.2.5. Let \( f : X \rightarrow Y \) be a map. Then \( f \) induces well-defined maps \( {\widetilde{f}}_{i} \) : \( {\widetilde{H}}_{i}\left( X\right) \rightarrow {\widetilde{H}}_{i}\left( Y\right) \) for every \( i \), where \( {\widetilde{f}}_{i} = {f}_{i} \mid {\widetilde{H}}_{i}\left( X\right) \) . | Proof. This follows immediately from the commutativity of the diagram\n\n | Yes |
Theorem 3.2.7. (1) Let \( A \) be a nonempty closed subset of \( X \). Suppose that \( \partial A \) has an open neighborhood \( C \) in \( A \) such that the inclusions \( \left( {X - A}\right) \cup C \rightarrow X - \operatorname{int}\left( A\right) \) and \( \partial A \rightarrow C \) are both homotopy equivalences... | Proof. (1) Let \( V = A - C \). Then \( V \) is a closed set in the interior of \( A \), so \( (X - V, A - V) \rightarrow \left( {X, A}\right) \) is excisive. But \( X - V = \left( {X - A}\right) \cup C \) and \( A - V = C \). By hypothesis the first of these is homotopy equivalent to \( X - \operatorname{int}\left( A\... | Yes |
Theorem 3.2.9. (1) Let \( A \) be a nonempty subset of \( X \) . Then for each \( i,{H}_{i}\left( {X, A}\right) \) is isomorphic to the reduced homology group \( {\widetilde{H}}_{i}\left( {X \cup {cA}}\right) \) . | Proof. (1) We follow the idea of the proof of Theorem 3.2.7. Let \( V = \{ \left( {a, s}\right) \mid s \geq \) \( \left. \frac{1}{2}\right\} \) so that \( V \) is a closed subset of \( X{ \cup }_{A}{cA} \) which is contained in the interior of \( {cA} \) . Then the inclusion \( \left( {\left( {X{ \cup }_{A}{cA}}\right)... | Yes |
Theorem 3.2.10 (Mayer-Vietoris). Let \( X = {X}_{1} \cup {X}_{2}, A = {X}_{1} \cap {X}_{2} \), and suppose that the inclusion \( \left( {{X}_{1}, A}\right) \rightarrow \left( {X,{X}_{2}}\right) \) is excisive. Then there is a long exact sequence in homology\n\n\[ \cdots \rightarrow {H}_{i}\left( A\right) \overset{\alph... | Proof. We have the long exact homology sequences\n\n\n\nwhere by assumption \( \varepsilon : {H}_{i}\left( {{X}_{1}, A}\right) \rightarrow {H}_{i}\left( {X,{X}_{2}}\right) \) is an isomorphism. Then the theorem follows... | Yes |
Theorem 3.2.11. Let \( \left( {X, A, B}\right) \) be a triad and suppose that the inclusion \( (A, A \cap \) \( B) \rightarrow \left( {A \cup B, B}\right) \) is excisive. Then there is an exact homology sequence\n\n\[ \cdots \rightarrow {H}_{i}\left( {X, A \cap B}\right) \rightarrow {H}_{i}\left( {X, A}\right) \oplus {... | Proof. Exactly the same as the proof of Theorem 3.2.10. | No |
Theorem 3.2.13. (1) For any space \( X \) there is an isomorphism, for any \( i \) , \[ \sum : {\widetilde{H}}_{i + 1}\left( {\sum X}\right) \rightarrow {\widetilde{H}}_{i}\left( X\right) \] | Proof. We prove (1). We have the exact homology sequence of the pair \( \left( {{c}_{ + }X, X}\right) \) : \[ \cdots \rightarrow {H}_{i + 1}\left( {{c}_{ + }X, X}\right) \rightarrow {H}_{i}\left( X\right) \rightarrow {H}_{i}\left( {{c}_{ + }X}\right) \rightarrow \cdots \] Now \( {c}_{ + }X \) is contractible to the con... | Yes |
Theorem 3.2.14. Let \( A \) and \( B \) be subspaces of \( X \) with \( B \subseteq A \) . Then there is an exact homology sequence\n\n\[ \cdots \rightarrow {H}_{i}\left( {A, B}\right) \rightarrow {H}_{i}\left( {X, B}\right) \rightarrow {H}_{i}\left( {X, A}\right) \overset{\partial }{ \rightarrow }{H}_{i - 1}\left( {A,... | Proof. We merely remark here that the boundary map in the sequence is the composition\n\n\[ {H}_{i}\left( {X, A}\right) \rightarrow {H}_{i - 1}\left( A\right) \rightarrow {H}_{i - 1}\left( {A, B}\right) \]\n\nOtherwise, the result follows directly from Theorem A.2.12. | No |
Theorem 3.2.15. Let \( X = {X}_{1} \cup {X}_{2}, A = {X}_{1} \cap {X}_{2} \), and suppose that the inclusion \( \left( {{X}_{1}, A}\right) \rightarrow \left( {X,{X}_{2}}\right) \) is excisive. Let \( B \) be an arbitrary subspace of \( A \) . Then there is a long exact sequence in homology | \[ \cdots \rightarrow {H}_{i}\left( {A, B}\right) \rightarrow {H}_{i}\left( {{X}_{1}, B}\right) \oplus {H}_{i}\left( {{X}_{2}, B}\right) \rightarrow {H}_{i}\left( {X, B}\right) \rightarrow {H}_{i - 1}\left( {A, B}\right) \rightarrow \cdots . \] | Yes |
Lemma 4.1.1. 1. \( {H}_{0}\left( {S}^{0}\right) \cong \mathbb{Z} \oplus \mathbb{Z} \) . More precisely, \( {H}_{0}\left( {S}^{0}\right) = \{ {mp} + {nq} \mid m, n \in \mathbb{Z}\} \) . | Proof. (1) Since \( \{ - 1\} \) and \( \{ 1\} \) are distinct components of \( {S}^{0} \), we have by Lemma 3.2.1 that \( {H}_{i}\left( {S}^{0}\right) \cong {H}_{i}\left( {\{ - 1\} }\right) \oplus {H}_{i}\left( {\{ 1\} }\right) \) . Since \( \{ - 1\} \) and \( \{ 1\} \) are both spaces consisting of a single point, the... | Yes |
Lemma 4.1.3. Fix a positive integer \( n \). 1. \( {H}_{n}\left( {S}^{n}\right) \cong \mathbb{Z} \) and \( {H}_{0}\left( {S}^{n}\right) \cong \mathbb{Z} \). 2. \( {\widetilde{H}}_{n}\left( {S}^{n}\right) \cong \mathbb{Z} \). 3. \( {H}_{i}\left( {S}^{n}\right) = 0 \) for \( i \neq 0, n \) and \( {\widetilde{H}}_{i}\left... | Proof. Observe that for any \( k,\sum {S}^{k} \) is homeomorphic to \( {S}^{k + 1} \). Then, if \( {\sum }^{i} \) denotes \( \sum \) applied \( i \) times, \( {\sum }^{i}{S}^{k} \) is homeomorphic to \( {S}^{k + i} \). In particular \( {\sum }^{n}{S}^{0} \) is homeomorphic to \( {S}^{n} \). Then, by repeated applicatio... | No |
Lemma 4.1.5. For any \( n \geq 1 \), there does not exist a retraction from \( {D}^{n} \) onto \( {S}^{n - 1} \) . | Proof. If there were such a retraction \( r : {D}^{n} \rightarrow {S}^{n - 1} \), then \( r \) would induce a surjection \( {r}_{i} : {H}_{i}\left( {D}^{n}\right) \rightarrow {H}_{i}\left( {S}^{n - 1}\right) \) for each \( i \), by Lemma 3.2.1(iii). But for \( i = n - 1,{H}_{n - 1}\left( {D}^{n}\right) = 0 \) and \( {H... | Yes |
Theorem 4.1.6 (Brouwer fixed-point theorem). Let \( f : {D}^{n} \rightarrow {D}^{n} \) be an arbitrary map. Then \( f \) has a fixed point, i.e. there is an \( {x}_{0} \in D \) with \( f\left( {x}_{0}\right) = {x}_{0} \) . | Proof. Suppose that \( f \) does not have a fixed point. Let \( r : {D}^{n} \rightarrow {S}^{n - 1} \) be the map defined as follows:\n\nFor \( x \in {D}^{n} \), take the line segment from \( f\left( x\right) \) to \( x \) and prolong it until it intersects \( {S}^{n - 1} \) at some point \( {x}^{\prime } \) . Then set... | Yes |
Theorem 4.1.7 (Invariance of domain). Let \( U \) be a nonempty open set in \( {\mathbb{R}}^{n} \) and \( V \) be a nonempty open set in \( {\mathbb{R}}^{m} \) and suppose there is a homeomorphism \( f : U \rightarrow V \) . Then \( m = n \) . | Proof. This is trivially true if \( m = 0 \) or \( n = 0 \), so we assume \( m \geq 1 \) and \( n \geq 1 \) . Although from a logical standpoint it is not necessary to begin with this special case, the basic idea of the proof comes through most clearly if we first consider the case \( U = {\mathbb{R}}^{m}, V = {\mathbb... | Yes |
Lemma 4.1.9. Let \( a : {S}^{n} \rightarrow {S}^{n} \) be the antipodal map, i.e., \( a\left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) = \) \( \left( {-{x}_{1},\ldots , - {x}_{n + 1}}\right) \) . Then the degree of a is \( {\left( -1\right) }^{n + 1} \) . | Proof. We divide the proof into two cases.\n\nCase \( 1 \) ( \( n \) is odd, \( n = {2m} - 1 \) ). Then we may regard \( {S}^{n} \) as the unit sphere in \( {\mathbb{C}}^{m} \) , and \( a : {S}^{n} \rightarrow {S}^{n} \) is \( a\left( {{z}_{1},\ldots ,{z}_{m}}\right) = \left( {-{z}_{1},\ldots , - {z}_{m}}\right) \) . B... | Yes |
(a) Let \( X = \left\lbrack {0,1}\right\rbrack \) and \( A = \{ 1\} \) . Let \( U = \{ 1\} \), a closed set. | Then \( U \subseteq A \) but the closure of \( U \) (i.e., \( U \) itself) is not contained in the interior of \( A \) . Now \( \left( {X - U, A - U}\right) = \left( {\lbrack 0,1}\right) ,\varnothing ) \) and \( \lbrack 0,1) \) is homotopy equivalent to a point, so in particular \( {H}_{0}\left( {X - U, A - U}\right) =... | Yes |
Lemma 4.2.7. A CW-complex \( X \) has the following properties:\n\n1. (Closure-finiteness) The closure of each cell in \( X \) intersects only finitely many other cells in \( X \) .\n\n2. (Weak topology) A subset A of \( X \) is closed if and only if the intersection of \( A \) with the closure of every cell in \( X \)... | Proof. (1) The closure of each cell is \( f\left( {D}_{\lambda }^{n}\right) \), the image of a compact set, and hence compact, and if (1) were false this set would have an infinite subset (one point from each other cell) without an accumulation point, which is impossible. | No |
Lemma 4.2.10. Let \( X \) be obtained from \( A \) by adjoining an \( n \) -cell. Then\n\n\[ \n{H}_{i}\left( {X, A}\right) = \left\{ \begin{array}{ll} \mathbb{Z} & i = n \\ 0 & i \neq n \end{array}\right.\n\] | Proof. Let \( C = \left\{ {x \in {D}^{n}\left| \right| x \mid \geq 1/2}\right\} \) . Then \( C \) is a \ | No |
Lemma 4.2.11. 1. \( {H}_{i}\left( {{X}^{n},{X}^{n - 1}}\right) = 0 \) for \( i \neq n \) . | Proof. This is just an elaboration of Lemma 4.2.10.\n\nLet \( \left( {{D}^{n}\left( \frac{1}{2}\right) ,{S}^{n - 1}\left( \frac{1}{2}\right) }\right) \) be the pair consisting of the disk of radius \( \frac{1}{2} \) and its boundary. Then the inclusions induce isomorphisms on homology\n\n\[ {H}_{ * }\left( {{D}^{n}\lef... | Yes |
Lemma 4.2.14. \( {C}_{ * }^{\text{cell }}\left( X\right) \) is a chain complex. | Proof. We need only check that \( {\partial }_{n - 1}{\partial }_{n} = 0 \) . But this is the composition\n\n\[ \n{H}_{n}\left( {{X}^{n},{X}^{n - 1}}\right) \overset{\partial }{ \rightarrow }{H}_{n - 1}\left( {X}^{n - 1}\right) \rightarrow {H}_{n - 1}\left( {{X}^{n - 1},{X}^{n - 2}}\right) \]\n\n\[ \n\overset{\partial ... | Yes |
Lemma 4.2.16. The group \( {C}_{n}^{\text{cell }}\left( X\right) \) is the free abelian group on the n-cells of \( X \) . If \( {\alpha }_{\lambda }^{n} \) is the generator corresponding to the n-cell \( {D}_{\lambda }^{n},\lambda \in {\Lambda }_{n} \), then \( \partial \left( {\alpha }_{\lambda }^{n}\right) \) is give... | Proof. This follows directly from Lemma 4.2.11 and its proof. | No |
Theorem 4.2.20. Let \( X \) be a finite CW-complex. Then\n\n\[ \chi \left( X\right) = \mathop{\sum }\limits_{i}{\left( -1\right) }^{i} \cdot \text{ number of }i\text{-cells of }X. \] | Proof. Let \( X \) have \( {d}_{i}i \) -cells and suppose \( {d}_{i} = 0 \) for \( i > n \) . We have the cellular chain complex of \( X \)\n\n\[ 0 \rightarrow {C}_{n}^{\mathrm{{cell}}}\left( X\right) \rightarrow {C}_{n - 1}^{\mathrm{{cell}}}\left( X\right) \rightarrow \cdots \rightarrow {C}_{1}^{\mathrm{{cell}}}\left(... | Yes |
Theorem 4.2.23. Let \( X \) be a finite \( {CW} \) -complex. Let \( \widetilde{X} \) be an \( n \) -fold cover of \( X \) . Then \( \chi \left( \widetilde{X}\right) = {n\chi }\left( X\right) \) . | Proof. Given any cell decomposition of \( X \), we may refine it to obtain a cell decomposition so that every cell is evenly covered by the covering projection. Then the inverse image of every cell is \( n \) cells, so the theorem immediately follows from Theorem 4.2.20. | Yes |
Let \( R \) be a \( k \) -leafed rose, and let \( \widetilde{R} \) be any \( n \) -fold cover of \( R \) . Then \( R \) has one 0 -cell and \( {k1} \) -cells, so \( \chi \left( R\right) = 1 - k \) (which of course agrees with \( {H}_{0}\left( R\right) = \mathbb{Z} \) and \( {H}_{1}\left( \mathbb{R}\right) = {\mathbb{Z}... | Now \( {H}_{0}\left( \widetilde{R}\right) = \mathbb{Z} \) (as by definition, a cover is connected), so we must have\n\n\[ 1 - \operatorname{rank}{H}_{1}\left( \widetilde{R}\right) = n\left( {1 - k}\right) \]\n\nand hence \( {H}_{1}\left( \widetilde{R}\right) = {\mathbb{Z}}^{n\left( {k - 1}\right) + 1} \) . (Compare Cor... | Yes |
Lemma 4.2.26. Let \( f : X \rightarrow Y \) be a cellular map. Then for each \( i, f \) induces a map \( {f}_{i}^{\text{cell }} : {H}_{i}^{\text{cell }}\left( X\right) \rightarrow {H}_{i}^{\text{cell }}\left( Y\right) . | Proof. By hypothesis, \( f \) induces a map \( {H}_{i}\left( {{X}^{n},{X}^{n - 1}}\right) \rightarrow {H}_{i}\left( {{Y}^{n},{Y}^{n - 1}}\right) \) for each \( i \) and \( n \), and then it is easy to check this induces a map on cellular homology. | No |
Theorem 4.2.27. Let \( X \) and \( Y \) be \( {CW} \) -complexes and let \( f : X \rightarrow Y \) be a cellular map. Then the following diagram commutes: | Proof. This follows easily from the commutativity of the diagram where the vertical maps are all induced by \( f \) . | No |
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