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Lemma 13.1. Let \( \mathcal{E} \) and \( \mathcal{F} \) be \( F \) -spaces, let \( T \) be a continuous linear transformation of \( \mathcal{E} \) onto \( \mathcal{F} \), and let \( \varepsilon \) be a positive number. Then \( {\left( T\left( {\mathcal{E}}_{\varepsilon }\right) \right) }^{ - } \) is a neighborhood of 0... | Proof. For each \( x \) in \( \mathcal{E} \) the mapping \( \lambda \rightarrow {\lambda x} \) is continuous on \( \mathbb{C} \) . Hence \( x/n \in {\mathcal{E}}_{\varepsilon /2} \) for all sufficiently large positive integers \( n \), and it follows that the sequence of sets \( {\left\{ n{\mathcal{E}}_{\varepsilon /2}... | Yes |
Lemma 13.2. Let \( \mathcal{E},\mathcal{F} \), and \( T \) be as in Lemma 13.1, and let \( V \) be a neighborhood of 0 in & Then \( T\left( V\right) \) is a neighborhood of 0 in \( \mathcal{F} \) . | Proof. Choose a positive number \( \varepsilon \) such that \( {\mathcal{E}}_{\varepsilon } \subset V \), and write \( {\varepsilon }_{n} = \varepsilon /{2}^{n} \) , \( n \in \mathbb{N} \) . According to Lemma 13.1, for each \( n \) there exists a positive number \( {\delta }_{n} \) such that\n\n\[ \n{\mathcal{F}}_{{\d... | Yes |
Theorem 13.3 (Open Mapping Theorem). If \( T \) is a continuous linear transformation of an \( F \) -space \( \mathcal{E} \) onto an \( F \) -space \( \mathcal{F} \), then \( T \) is an open mapping of \( \mathcal{E} \) onto \( \mathcal{F} \). In particular, if \( T \) is one-to-one, then the inverse transformation \( ... | Proof. It suffices to prove that \( T \) is open. Let \( U \) be an open set in \( \mathcal{E} \), and suppose \( {y}_{0} \) is a vector belonging to \( T\left( U\right) \). There exists a vector \( {x}_{0} \) in \( U \) such that \( T{x}_{0} = {y}_{0} \), and since \( U \) is open, \( {x}_{0} \) is an interior point o... | Yes |
Theorem 13.4 (Closed Graph Theorem). If \( T \) is a linear transformation of an \( F \) -space \( \mathcal{E} \) into an \( F \) -space \( \mathcal{F} \), then the graph of \( T \) is closed in \( \mathcal{E}{ \oplus }_{1}\mathcal{F} \) if and only if \( T \) is continuous. | Proof. The sufficiency of the condition is easily established. To see this, let \( T \) be continuous, and let \( \left( {{x}_{0},{y}_{0}}\right) \) be a vector in \( \mathcal{E}{ \oplus }_{1}\mathcal{F} \) that does not belong to \( \mathcal{G} \), so that \( {y}_{0} \neq T{x}_{0} \) . Then there exist disjoint open s... | Yes |
Corollary 13.5. Let \( T \) be a linear transformation of an \( F \) -space \( \mathcal{E} \) into a quasi-normed space \( \mathcal{F} \), and suppose that for every sequence \( \left\{ {z}_{n}\right\} \) in \( \mathcal{E} \) such that \( {z}_{n} \rightarrow 0 \) and such that the sequence \( \left\{ {T{z}_{n}}\right\}... | Proof. Consider the graph \( \mathcal{G} \) of \( T \) as a linear submanifold of \( \mathcal{E}{ \oplus }_{1}\widehat{\mathcal{F}} \) , where \( \widehat{\mathcal{F}} \) denotes the completion of \( \mathcal{F} \) (Th. 11.16). If \( \left\{ \left( {{x}_{n}, T{x}_{n}}\right) \right\} \) is a sequence in \( \mathcal{G} ... | Yes |
Theorem 13.6. Let \( \mathcal{E} \) and \( \mathcal{F} \) be quasinormed spaces, and let \( T \) be a linear transformation of \( \mathcal{E} \) into \( \mathcal{F} \) . Then any two of the following three conditions imply the third:\n\n(i) & is complete,\n\n(ii) \( T \) is continuous,\n\n(iii) \( T \) is closed. | Proof. That (ii) and (iii) imply one another in the presence of (i) is, once again, essentially the content of the closed graph theorem. Thus it need only be shown that if a quasinormed space \( \mathcal{E} \) is the domain of a linear transformation that is both closed and continuous, then \( \mathcal{E} \) must be co... | Yes |
Proposition 13.7. Let \( \mathcal{E} \) be a Banach space, let \( T \) be a linear transformation of \( \mathcal{E} \) into a normed space \( \mathcal{F} \), and suppose the function \( x \rightarrow \parallel {Tx}\parallel \) is lower semicontinuous on \( \mathcal{E} \) (Ex. \( 3\mathrm{\;K} \) ). Then \( T \) is boun... | Proof. The sets \( {M}_{n} = \{ x \in \mathcal{E} : \parallel {Tx}\parallel \leq n\} \) are closed for all positive integers \( n \) and cover \( \mathcal{E} \) . Hence, by the Baire category theorem, at least one \( {M}_{n} \) must have nonempty interior, and \( T \) is bounded on that open set. But then \( T \) is bo... | Yes |
Lemma 14.1. Let \( \mathcal{E} \) be a real normed space, and let \( \mathcal{M} \) be a linear manifold in \( \mathcal{E} \). Let \( f \) be a bounded linear functional on \( \mathcal{M} \), and suppose that \( {x}_{0} \) is a vector in 6 that does not belong to \( \mathcal{M} \). If \( {\mathcal{M}}_{0} \) denotes th... | Proof. It is clear that we may assume, without loss of generality, that \( \parallel f\parallel = 1 \). The linear manifold \( {\mathcal{M}}_{0} \) consists of all vectors of the form \( y + t{x}_{0} \), where \( y \) is an arbitrary vector in \( \mathcal{M} \) and \( t \) is a real number. If \( r \) denotes a real nu... | Yes |
Theorem 14.2 (Real Hahn-Banach Theorem). Let \( \mathcal{M} \) be a linear submanifold of a real normed space \( \mathcal{E} \), and let \( f \) be a bounded linear functional defined on \( \mathcal{M} \) . Then there exists a linear extension \( {f}_{0} \) off to the entire space &satisfying the condition \( \begin{Vm... | Proof. Let \( \mathcal{P} \) denote the collection of all pairs \( \left( {g,\mathcal{G}}\right) \), where \( \mathcal{G} \) is a linear submanifold of \( \mathcal{E} \) containing \( \mathcal{M} \) and \( g \) is a linear functional on \( \mathcal{G} \) that extends \( f \) and satisfies \( \parallel g\parallel = \par... | Yes |
Proposition 14.4. Let \( \mathcal{E} \) be a linear space equipped with a pseudonorm \( \sigma \), and let \( f \) be a linear functional defined on a linear submanifold \( \mathcal{M} \) of \( \mathcal{E} \) such that \( f \) is bounded on \( \mathcal{M} \) with respect to \( \sigma \) . Then there exists a linear ext... | Proof. We employ Proposition 11.17. If \( \mathcal{Z} = \{ z \in \mathcal{E} : \sigma \left( z\right) = 0\} \) denotes the zero space of \( \sigma \), then it is clear that \( f \) must vanish on \( \mathcal{M} \cap \mathcal{Z} \) . It follows at once that if we write \( \widetilde{\mathcal{M}} = \mathcal{M} + \mathcal... | Yes |
Proposition 14.5. Let \( {\left\{ {\sigma }_{\gamma }\right\} }_{\gamma \in \Gamma } \) be an indexed family of pseudonorms on a linear space & , let \( \mathcal{M} \) be a linear submanifold of &, and let \( f \) be a linear functional on \( \mathcal{M} \) that is continuous with respect to the linear topology induced... | Proof. By Problem 12U there exist indices \( {\gamma }_{1},\ldots ,{\gamma }_{n} \) and a positive number \( \varepsilon \) such that if \( x \in \mathcal{M} \) and if \( {\sigma }_{{\gamma }_{i}}\left( x\right) \leq \varepsilon, i = 1,\ldots, n \), then \( \left| {f\left( x\right) }\right| \leq 1 \) . If we set \( \si... | Yes |
Proposition 14.6. Let \( \mathcal{E} \) be a real linear space, and suppose given a real-valued function \( p \) on \( \mathcal{E} \) satisfying the following conditions:\n\n(i) \( p\left( {x + y}\right) \leq p\left( x\right) + p\left( y\right) \) for all \( x \) and \( y \) in \( \mathcal{E} \),\n\n(ii) \( p\left( {tx... | Sketch of proof of Proposition 14.6. We may suppose that \( \mathcal{M} \neq \mathcal{E} \). Let \( {x}_{0} \) be a vector in \( \mathcal{E} \) that does not belong to \( \mathcal{M} \), and, as in Lemma 14.1, let \( {\mathcal{M}}_{0} \) denote the linear submanifold of \( \mathcal{E} \) generated by \( \mathcal{M} \) ... | No |
Proposition 14.7. In any topological linear space & the following conditions are equivalent:\n\n(i) The topology on \( \mathcal{E} \) is induced by a family of pseudonorms,\n\n(ii) There is a base for the topology on & consisting of convex sets,\n\n(iii) There is a neighborhood base at the origin in & consisting of con... | Proof. If \( \sigma \) is a pseudonorm on \( \mathcal{E} \) then every ball \( \left\{ {x \in \mathcal{E} : \sigma \left( {x - {x}_{0}}\right) < r}\right\} \) is convex, and it follows at once that (i) implies (ii). Moreover, it is clear that (ii) implies (iii). To see that (iii) implies (iv), let \( C \) be a convex n... | Yes |
Proposition 14.8. In any topological linear space & the following conditions are equivalent:\n\n(i) The topology on \( \mathcal{E} \) is induced by a countably determined separating family of pseudonorms,\n\n(ii) The topology on \( \mathcal{E} \) is induced by a countable separating family of pseudo-norms,\n\n(iii) & i... | Proof. If the given topology on \( \mathcal{E} \) is induced by some countably determined separating family \( {\left\{ {\sigma }_{\gamma }\right\} }_{\gamma \in \Gamma } \) of pseudonorms, and if \( {\left\{ {\sigma }_{\gamma }\right\} }_{\gamma \in {\Gamma }_{0}} \) is a countable cofinal subfamily, then \( {\left\{ ... | Yes |
Proposition 14.9. Let \( \mathcal{E} \) be a linear space and let \( {\left\{ {\sigma }_{n}\right\} }_{n = 1}^{\infty } \) be a separating sequence of pseudonorms on \( \mathcal{E} \) . Then \( \mathcal{E} \) is a Frechét space in the topology induced by the sequence \( \left\{ {\sigma }_{n}\right\} \) if and only if e... | Proof. The quasinorm | | defined in (16) induces the same topology on \( \mathcal{E} \) as does the sequence \( \left\{ {\sigma }_{n}\right\} \) . Hence a sequence \( {\left\{ {x}_{p}\right\} }_{p = 1}^{\infty } \) in \( \mathcal{E} \) converges to a limit \( x \) with respect to \( \parallel \) if and only if it conve... | Yes |
Proposition 14.10. Let \( \mathcal{M} \) be a (closed) subspace of a normed space \( \mathcal{E} \), and suppose \( {x}_{0} \) is a vector in &that does not belong to \( \mathcal{M} \). Then there exists a functional fin \( {\mathcal{E}}^{ * } \) satisfying the conditions:\n\n(i) \( f\left( \mathcal{M}\right) = \left( ... | Proof. Let \( {\mathcal{M}}_{0} \) denote the linear submanifold of \( \mathcal{E} \) generated by \( \mathcal{M} \) and \( {x}_{0} \). If we define a linear functional \( g \) on \( {\mathcal{M}}_{0} \) by setting\n\n\[ g\left( {y + \lambda {x}_{0}}\right) = \lambda \]\n\nfor all vectors \( y \) in \( \mathcal{M} \) a... | Yes |
Let \( \mathcal{E} \) be a locally convex space and let \( \mathcal{M} \) be a linear sub-manifold of \( \mathcal{E} \). For each linear functional \( f \) in \( {\mathcal{M}}^{ * } \) there exists a linear functional \( {f}_{0} \) in \( {\mathcal{E}}^{ * } \) such that \( {f}_{0} \mid \mathcal{M} = f \). Moreover, if ... | Proof. By Proposition 14.7 the given topology on \( \mathcal{E} \) is induced by a family \( \left\{ {\sigma }_{\gamma }\right\} \) of pseudonorms. Hence the first assertion is an immediate consequence of Proposition 14.5. Moreover, if \( \mathcal{M} \) is closed and \( {x}_{0} \notin \mathcal{M} \), then there exist i... | Yes |
Proposition 14.14. Let \( \mathcal{E} \) be a topological linear space, and let \( C \) and \( D \) be nonempty disjoint convex subsets of \( \mathcal{E} \), one of which at least has nonempty interior. Then there exists a hyperplane separating \( C \) and \( D \) . | Proof. Since an interior point of a convex set is certainly an internal point in the sense of Example \( \mathrm{F} \), it follows at once from Example \( \mathrm{G} \) that there exists a nonzero real linear functional \( f \) on \( \mathcal{E} \) such that \( f\left( u\right) \leq f\left( v\right) \) whenever \( u \i... | Yes |
Proposition 14.15. Let \( \mathcal{E} \) be a locally convex topological linear space, let \( C \) be a closed convex set in \( \mathcal{E} \), and let \( {x}_{0} \) be a vector in \( \mathcal{E} \) that does not belong to C. Then there exists a linear functional \( g \) in \( {\mathcal{E}}^{ * } \) and a real number \... | Proof. Let \( V \) be a convex open neighborhood of \( {x}_{0} \) that does not meet \( C \) . Then, as was seen in the proof of Proposition 14.14, there exists a continuous nonzero real linear functional \( f \) on \( \mathcal{E} \) such that \( f\left( u\right) \leq f\left( v\right) \) whenever \( u \in C \) and \( v... | Yes |
For any normed space \( \mathcal{E} \) and for each \( f \) in \( {\mathcal{E}}^{ * } \) the function \( {\sigma }_{f} \) defined in (1) is a pseudonorm on & Moreover, if T denotes an arbitrary linear topology on \( \mathcal{E} \), then \( f \) is continuous with respect to \( \mathcal{T} \) if and only if \( {\sigma }... | The verification that \( {\sigma }_{f} \) is a pseudonorm is routine and is left to the reader (cf. Example 11M). Suppose that \( f \) is continuous with respect to some topology \( \mathcal{T} \) on \( \mathcal{E} \) and let \( {x}_{0} \) be a vector in \( \mathcal{E} \) . Then for any given \( \varepsilon > 0 \) ther... | No |
Proposition 15.2. The weak topology on a normed space & is a linear topology on & turning & into a separated locally convex space. The weak topology coincides with the norm topology when and only when & is finite dimensional. | Proof. The weak topology on \( \mathcal{E} \) coincides with the (linear) topology induced on \( \mathcal{E} \) by the family of pseudonorms \( {\left\{ {\sigma }_{f}\right\} }_{f \in {\mathcal{E}}^{ * }} \), and is therefore locally convex (Prop. 14.7). To see that the weak topology is also Hausdorff, note that \( \ma... | Yes |
Proposition 15.3. In any normed space & the collection \( \mathcal{V} \) of all sets of the form (2) constitutes a base of open neighborhoods of 0 in the weak topology on \( \mathcal{E} \) ; the collection of all translates\n\n\[ \n{x}_{0} + U\left( {{f}_{1},\ldots ,{f}_{n};\varepsilon }\right) = \left\{ {x \in \mathca... | Proof. It suffices to show that \( \mathcal{V} \) is a base of open neighborhoods of 0 in the weak topology (Prob. 11L(ii)). To see this we note that \( U\left( {{f}_{1},\ldots ,{f}_{n};\varepsilon }\right) = \)\n\n\( U\left( {{f}_{1};\varepsilon }\right) \cap \cdots \cap U\left( {{f}_{n};\varepsilon }\right) \) is the... | No |
Proposition 15.4. A net \( \left\{ {x}_{\lambda }\right\} \) in a normed space &converges to a limit \( {x}_{0} \) in the weak topology on \( \mathcal{E} \) if and only if the net \( \left\{ {f\left( {x}_{\lambda }\right) }\right\} \) converges to \( f\left( {x}_{0}\right) \) for every \( f \) in \( {\mathcal{E}}^{ * }... | Proof. According to Proposition 11.27, a net \( \left\{ {x}_{\lambda }\right\} \) in \( \mathcal{E} \) converges to \( {x}_{0} \) with respect to the weak topology on \( \mathcal{E} \) if and only if \( \lim {\sigma }_{f}\left( {{x}_{0} - {x}_{\lambda }}\right) = 0 \) for every \( f \) in \( {\mathcal{E}}^{ * } \), i.e... | Yes |
Let \( \mathcal{F} \) be a normed space and let \( {\mathcal{F}}_{w} \) denote the topological linear space obtained by equipping \( \mathcal{F} \) with its weak topology. Let \( \mathcal{E} \) be an arbitrary topological linear space, and let \( T \) be a linear transformation of \( \mathcal{E} \) into \( \mathcal{F} ... | It is enough to show that the condition is sufficient. Let \( \left\{ {x}_{\lambda }\right\} \) be a net in \( \mathcal{E} \) such that \( \mathop{\lim }\limits_{\lambda }{x}_{\lambda } = 0 \) . Then the net \( \left\{ {f\left( {T{x}_{\lambda }}\right) }\right\} \) converges to zero in \( \mathbb{C} \) for each \( f \)... | No |
Proposition 15.6. If 6 is an arbitrary normed space, then the linear transformation \( j \) of \( \mathcal{E} \) into \( {\mathcal{E}}^{* * } \) defined by setting\n\n\[ j\left( x\right) \left( f\right) = f\left( x\right) ,\;f \in {\mathcal{E}}^{ * },\;x \in \mathcal{E}, \]\n\nis an isometry. | Proof. Let \( x \) be a vector in \( \mathcal{E} \) . By Corollary 14.11 there exists a linear functional \( {f}_{0} \) in \( {\mathcal{E}}^{ * } \) such that \( \begin{Vmatrix}{f}_{0}\end{Vmatrix} = 1 \) and such that \( j\left( x\right) \left( {f}_{0}\right) = {f}_{0}\left( x\right) = \parallel x\parallel \) . Thus \... | No |
Proposition 15.7. A set \( M \) in a normed space \( \mathcal{E} \) is weakly bounded if and only if it is bounded (in norm). | Proof. It is clear that if \( M \) is bounded, then \( M \) is weakly bounded. On the other hand, if \( M \) is weakly bounded, and if \( j \) denotes the natural embedding of \( \mathcal{E} \) in \( {\mathcal{E}}^{* * } \), then \( j\left( M\right) \) is, by hypothesis, pointwise bounded on \( {\mathcal{E}}^{ * } \), ... | Yes |
Proposition 15.9. Let \( \mathcal{E} \) be a linear space, let \( \mathcal{M} \) be a linear manifold of linear functionals on \( \mathcal{E} \), and let \( \mathcal{F} \) be a linear submanifold of \( \mathcal{E} \) that is closed with respect to the topology induced on \( \mathcal{E} \) by \( \mathcal{M} \). Then for... | Proof. As noted above, the topology induced on \( \mathcal{E} \) by \( \mathcal{M} \) is the same as that induced by the family of pseudonorms \( {\left\{ {\sigma }_{f}\right\} }_{f \in \mathcal{M}} \), and is therefore locally convex. Hence Proposition 14.13 applies, and since every linear functional on \( \mathcal{E}... | No |
Proposition 15.10. For any normed space & the weak* topology on & turns \( {\mathcal{E}}^{ * } \) into a separated, locally convex, topological linear space. A net \( \left\{ {f}_{\lambda }\right\} \) in \( {\mathcal{E}}^{ * } \) converges to a limit \( f \) in the weak \( {}^{ * } \) topology if and only if \( \mathop... | Proof. It is obvious that \( j\left( \mathcal{E}\right) \) is separating on \( {\mathcal{E}}^{ * } \) . Everything else follows from Proposition 15.8. | No |
Proposition 15.12. If \( {\left\{ {\mathcal{T}}_{\gamma }\right\} }_{\gamma \in \Gamma } \) is an arbitrary indexed family of locally convex linear topologies on a linear space \( \mathcal{E} \), then \( \mathop{\sup }\limits_{\gamma }{\mathcal{T}}_{\gamma } \) is also a locally convex linear topology on \( \mathcal{E}... | Proof. It suffices to show that the topology sup, \( {\mathcal{T}}_{\gamma } \) is locally convex (Prop. 11.25). If \( {V}_{i} = {V}_{{\gamma }_{i}} \) is an absolutely convex neighborhood of 0 in \( \mathcal{E} \) with respect to \( {\mathcal{T}}_{{\gamma }_{i}}, i = 1,\ldots, n \), then \( {V}_{1} \cap \cdots \cap {V... | Yes |
Proposition 15.13. Suppose \( \mathcal{E} \) is the inductive limit of a net of linear sub-manifolds \( \left\{ {\mathcal{M}}_{\lambda }\right\} \) . Then a pseudonorm \( \sigma \) on \( \mathcal{E} \) is continuous if and only if the restriction of \( \sigma \) to each \( {\mathcal{M}}_{\lambda } \) is continuous with... | Proof. A pseudonorm \( \sigma \) on \( \mathcal{E} \) is continuous with respect to the inductive limit topology if and only if the topology \( {\mathcal{T}}_{\sigma } \) induced by \( \sigma \) is refined by that topology, i.e., if and only if the inclusion mappings \( {i}_{\lambda } : {\mathcal{M}}_{\lambda } \righta... | Yes |
Proposition 16.8. Let \( \mathcal{E} \) and \( \mathcal{F} \) be separated locally convex topological linear spaces, and let \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) denote the linear space of all continuous linear transformations of \( \mathcal{E} \) into \( \mathcal{F} \) . Then the mapping \( T \righ... | Proof. The verification that the mapping \( T \rightarrow {T}^{ * } \) is a linear transformation is routine, and will be left to the reader. To show that it is an isomorphism of \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) into the space of all linear transformations of \( {\mathcal{F}}^{ * } \) into \( {\... | No |
Proposition 16.9. If \( \mathcal{E} \) and \( \mathcal{F} \) are Banach spaces, then the mapping that assigns to each bounded linear transformation \( T \) in \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) its adjoint \( {T}^{ * } \) is an isometric isomorphism of \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}... | Proof. All that is needed is to prove that \( {T}^{ * } \) is bounded and \( \begin{Vmatrix}{T}^{ * }\end{Vmatrix} = \parallel T\parallel \) for every \( T \) in \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \), for everything else has already been established. Moreover, since \( \begin{Vmatrix}{\left( {{T}^{ *... | Yes |
Corollary 16.11. If \( T \) is an isometric isomorphism of \( \mathcal{E} \) onto \( \mathcal{F} \), then \( {T}^{ * } \) is an isometric isomorphism of \( {\mathcal{F}}^{ * } \) onto \( {\mathcal{E}}^{ * } \) . | Proof. Since \( T \) is an isometric isomorphism we have \( \parallel T\parallel = \begin{Vmatrix}{T}^{-1}\end{Vmatrix} = 1 \) , from which it follows, according to Proposition 16.9 and Corollary 16.10, that \( \begin{Vmatrix}{T}^{ * }\end{Vmatrix} = \begin{Vmatrix}{\left( {T}^{ * }\right) }^{-1}\end{Vmatrix} = 1 \) . ... | Yes |
Proposition 16.14. If \( \langle \mathcal{E},\mathcal{F}\rangle \) is a dual pair and if \( M \) is a subset of \( \mathcal{E} \), then the polar \( {M}^{0} \) is an absolutely convex subset of \( \mathcal{F} \) that is closed in the topology induced on \( \mathcal{F} \) by \( \mathcal{E} \), and the prepolar \( {}^{0}... | The proof of Proposition 16.14 is so like that of Proposition 16.2 in every particular that we omit it. The trick is to recall Problem 15J and use the following lemma in place of Proposition 14.13. | No |
Lemma 16.15. Let & be a separated locally convex topological linear space. If \( A \) is a nonempty, closed, absolutely convex subset of \( \mathcal{E} \) and if \( {x}_{0} \) is a vector in \( \mathcal{E} \) not belonging to \( A \), then there exists a functional \( {f}_{0} \) in \( {\mathcal{E}}^{ * } \) such that \... | Proof. Since \( {x}_{0} \) is a vector not belonging to \( A \), then by Proposition 14.15 there exists a functional \( g \) in \( {\mathcal{E}}^{ * } \) and a real number \( c \) such that\n\n\[ \operatorname{Re}g\left( x\right) \leq c, x \in A\text{, while}\operatorname{Re}g\left( {x}_{0}\right) > c\text{.} \]\n\n(1)... | Yes |
Proposition 16.17. Let \( \langle \mathcal{E},\mathcal{F}\rangle \) be a dual pair. If \( M \) is a subset of \( \mathcal{E} \) that is bounded in the topology induced on & by F, then the polar \( {M}^{0} \) is absorbing. Dually, if \( A \) is an absorbing subset of \( \mathcal{E} \), then \( {A}^{0} \) is bounded with... | Proof. If \( \mathop{\sup }\limits_{{x \in M}}\left| {\langle x, y\rangle }\right| = K < + \infty \), then \( {\lambda y} \in {M}^{0} \) whenever \( \left| \lambda \right| \leq 1/K \) . Thus \( {M}^{0} \) is absorbing whenever \( M \) is bounded in the topology induced by \( \mathcal{F} \) . Similarly, if \( A \) is ab... | Yes |
If \( \langle \mathcal{E},\mathcal{F}\rangle \) is a dual pair, then a subset \( M \) of \( \mathcal{E} \) is bounded in the topology induced on & by F if and only if \( {M}^{0} \) is absorbing. Likewise if \( A \) is a subset of \( \mathcal{E} \) that is absolutely convex and closed in the topology induced by \( \math... | It suffices to give the half of the proof that has not already been given. If \( M \) is a subset of \( \mathcal{E} \) such that \( {M}^{0} \) is absorbing, then \( {}^{0}\left( {M}^{0}\right) \) is bounded in the topology induced by \( \mathcal{F} \) (simply apply the foregoing result to the reversed pair \( \langle \... | No |
Proposition 16.19. Let \( \langle \mathcal{E},\mathcal{F}\rangle \) be a dual pair, and let \( \mathcal{A} \) be an admissible collection of weakly bounded subsets of \( \mathcal{E} \) . Then the topology \( \mathcal{T} \) of uniform convergence on the sets in \( \mathcal{A} \) is a separated locally convex linear topo... | Proof. The topology \( \mathcal{T} \) is automatically locally convex, being induced by a family of pseudonorms. That \( \mathcal{T} \) is separated follows at once from condition (iii) in the foregoing definition. Moreover, condition (i) implies that the family of pseudonorms inducing \( \mathcal{T} \) is saturated, s... | Yes |
Proposition 16.20. If \( \mathcal{T} \) is a topology of a dual pair \( \langle \mathcal{E},\mathcal{F}\rangle \), then the collection \( \mathcal{A} \) of all those subsets of \( \mathcal{F} \) that are equicontinuous with respect to \( \mathcal{T} \) is an admissible collection of weakly bounded subsets of \( \mathca... | Proof. It is obvious that the topologies of uniform convergence on the polars of any two neighborhood bases at 0 in \( \mathcal{E} \) coincide (Ex. U). Let \( \mathcal{V} \) be a neighborhood base at 0 in \( \mathcal{E} \) with respect to \( \mathcal{T} \) that consists exclusively of closed absolutely convex sets (cf.... | Yes |
Lemma 16.21. Let \( \mathcal{E} \) be a separated locally convex space with dual \( {\mathcal{E}}^{ * } \), and let A be an absolutely convex compact subset of & Then every linear functional on \( \mathcal{E} * \) that is bounded on the polar \( {A}^{0} \) is determined by a vector \( x \) in \( \mathcal{E} \) (that is... | Proof. Let \( {\mathcal{E}}^{*\prime } \) denote the full algebraic dual of \( {\mathcal{E}}^{ * } \), and let \( k \) denote the natural mapping of \( \mathcal{E} \) into \( {\mathcal{E}}^{*\prime } \) . Then \( k \) is a linear space isomorphism of \( \mathcal{E} \) into \( {\mathcal{E}}^{*\prime } \) and is also cle... | Yes |
Theorem 16.22 (Mackey-Arens Theorem). A locally convex linear topology \( \mathcal{T} \) on the space \( \mathcal{E} \) of a dual pair \( \langle \mathcal{E},\mathcal{F}\rangle \) is a topology of that dual pair if and only if it is the topology of uniform convergence on the sets of some admissible collection of absolu... | Proof. Suppose first that the topology \( \mathcal{T} \) is induced by such a collection of sets, and let \( \varphi \) be a linear functional on \( \mathcal{E} \) that is continuous with respect to \( \mathcal{T} \) . Then, since \( \mathcal{A} \) is admissible, there exists a set \( A \) in \( \mathcal{A} \) such tha... | Yes |
Theorem 16.23 (Alaoglu-Bourbaki Theorem). If \( \mathcal{E} \) is a separated locally convex space and if \( V \) is an arbitrary neighborhood of the origin in \( \mathcal{E} \) , then the polar \( {V}^{0} \) is weak* compact. | Proof. Since \( V \) is a neighborhood of 0 it is absorbing. For each vector \( x \) in \( \mathcal{E} \) let \( t\left( x\right) \) be a positive number large enough so that \( x \in t\left( x\right) V \) (in particular, set \( t\left( x\right) = 1 \) for each \( x \) in \( V \) ), and form the product\n\n\[ \Pi = \ma... | "No" |
Theorem 16.24 (Šmul’yan Criterion). A normed space & is reflexive if and only if its closed unit ball \( {\mathcal{E}}_{1} \) is weakly compact. | Proof. As was noted in the proof of Proposition 16.2 in a somewhat more general context, it is obvious that the natural embedding \( j \) of \( \mathcal{E} \) in \( {\mathcal{E}}^{* * } \) is a homeomorphism between \( \mathcal{E} \) in its weak topology and \( j\left( \mathcal{E}\right) \) in the (relative) weak* topo... | Yes |
Proposition 17.1. If \( \left( {X,\mathbf{S},\mu }\right) \) is a measure space and if \( p \) is an arbitrary positive number, then the collection \( {\mathcal{L}}_{p}\left( X\right) = {\mathcal{L}}_{p}\left( {X,\mathbf{S},\mu }\right) \) of all those measurable complex-valued functions fon \( \left( {X,\mathbf{S}}\ri... | Proof. That \( {\mathcal{L}}_{p}\left( X\right) \) is a linear space for every value of \( p \) follows from the elementary inequality \( {\left( u + v\right) }^{p} \leq {2}^{p}\left( {{u}^{p} + {v}^{p}}\right) \), valid for \( u, v \geq 0 \) and all positive \( p \) . To show that the function defined in (1) is a pseu... | Yes |
Lemma 17.3. Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space and, for an arbitrary \( p \) , \( 1 \leq p \leq + \infty \), let \( \left\{ {g}_{n}\right\} \) be a sequence of functions in \( {\mathcal{L}}_{p}\left( X\right) \) with the property that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\begin{Vm... | Proof. The case \( p = + \infty \) is trivial and will be omitted. For \( 1 \leq p < + \infty \) , we note first that it suffices to treat the case in which the functions \( {g}_{n} \) are all nonnegative (for we may simply replace \( {g}_{n} \) by \( \left| {g}_{n}\right| \) ). Set \( {h}_{n} = {g}_{1} + \cdots + {g}_... | Yes |
For any measure space \( \left( {X,\mathbf{S},\mu }\right) \) the Lebesgue spaces \( {\mathcal{L}}_{p}\left( {X,\mathbf{S},\mu }\right) \), \( 1 \leq p \leq + \infty \), are complete, and are therefore Banach spaces. | It suffices as always (see Problem 4E) to show that if a Cauchy sequence \( \left\{ {f}_{n}\right\} \) satisfies the added condition\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\begin{Vmatrix}{f}_{n + 1} - {f}_{n}\end{Vmatrix}}_{p} < + \infty \]\n\n(4)\n\nthen there exists a function \( f \) in \( {\mathcal{L}}_{p}... | Yes |
Theorem 17.6. If \( \left( {X,\mathbf{S},\mu }\right) \) is a \( \sigma \) -finite measure space, then \( {\mathcal{L}}_{1}{\left( X,\mathbf{S},\mu \right) }^{ * } \) is isometrically isomorphic to \( {\mathcal{L}}_{\infty }\left( {X,\mathbf{S},\mu }\right) \) under the correspondence \( {\varphi }_{g} \rightarrow g \)... | Proof. Clearly if \( g \) is essentially bounded and \( f \) is integrable, then\n\n\[ \left| {{\int }_{X}{fgd\mu }}\right| \leq \parallel g{\parallel }_{\infty }\parallel f{\parallel }_{1} \]\n\nso that \( \begin{Vmatrix}{\varphi }_{q}\end{Vmatrix} \leq \parallel g{\parallel }_{\infty } \) . Hence, as before, it suffi... | Yes |
Theorem 17.7. For any measure space \( \left( {X,\mathbf{S},\mu }\right) \) the following conditions are equivalent:\n\n(i) \( {\mathcal{L}}_{p}\left( X\right) \) is separable for some one \( p,1 \leq p < + \infty \) ,\n\n(ii) \( \left( {X,\mathbf{S},\mu }\right) \) is separable,\n\n(iii) \( {\mathcal{L}}_{p}\left( X\r... | Proof. The collection \( S \) of all characteristic functions of sets of finite measure is a subset of \( {\mathcal{L}}_{p}\left( X\right) \) . Consequently, if \( {\mathcal{L}}_{p}\left( X\right) \) is separable for some one value of \( p \), then \( S \) is also separable in the relative metric. (Recall (Prop. 4.1) t... | No |
Lemma 17.8. Let \( \left( {X,\mathbf{S}}\right) \) be a measurable space, let \( \mathcal{E} \) be a normed space, and let \( \Phi \) be a measurable \( \mathcal{E} \) -valued mapping on \( X \) . Then the function\n\n\[ \n{N}_{\Phi }\left( x\right) = \parallel \Phi \left( x\right) \parallel ,\;x \in X,\n\]\n\n(6)\n\ni... | Proof. The set \( \{ x \in X : \parallel \Phi \left( x\right) \parallel \leq r\} \) can be written as \( \left\{ {x \in X : \Phi \left( x\right) \in {\mathcal{E}}_{r}}\right\} \) , \( r > 0 \), and the closed balls \( {\mathcal{E}}_{r} \) are Borel sets in \( \mathcal{E} \) . (Recall Problem 6C.) | Yes |
Proposition 17.9. Let \( \left( {X,\mathbf{S},\mu }\right) \) be a measure space, let \( \mathcal{E} \) be a normed space, and let \( p \) be a positive real number. Then the collection \( {\mathcal{L}}_{p}\left( {X;\mathcal{E}}\right) = \) \( {\mathcal{L}}_{p}\left( {X,\mathbf{S},\mu ;\mathcal{E}}\right) \) of all tho... | Proof. If \( \Phi \) and \( \Psi \) are arbitrary \( \mathcal{E} \) -valued mappings, then \( {N}_{\Phi + \Psi } \leq {N}_{\Phi } + {N}_{\Psi } \) by the triangle inequality in \( \mathcal{E} \), and since \( {\mathcal{L}}_{p}\left( X\right) \) is a linear space, it follows at once that \( {\mathcal{L}}_{p}\left( {X;\m... | Yes |
Proposition 17.13. If \( \left( {X,\mathbf{S},\mu }\right) \) is a measure space and &is a separable Banach space, then the integrable simple \( \mathcal{E} \) -valued mappings on \( X \) constitute a dense linear manifold in \( {\mathcal{L}}_{p}\left( {X;\mathcal{E}}\right) \) for \( 1 \leq p < + \infty \) . Furthermo... | Proof. That the collection \( \mathcal{S} \) of integrable simple \( \mathcal{E} \) -valued mappings constitutes a linear submanifold of \( {\mathcal{L}}_{p}\left( {X;\mathcal{E}}\right) \) for all values of \( p \) is obvious. If \( \Phi \) belongs to \( {\mathcal{L}}_{p}\left( {X;\mathcal{E}}\right) \) then the funct... | Yes |
Theorem 17.14. Let \( \\left( {X,\\mathbf{S},\\mu }\\right) \) be a measure space, and let \( \\mathcal{E} \) be a separable Banach space. Then there exists a unique bounded linear transformation \( L \) of \( {\\mathcal{L}}_{1}\\left( {X;\\mathcal{E}}\\right) \) into \( \\mathcal{E} \) satisfying the condition\n\n\[ L... | Proof. Suppose first that \( T \) is an arbitrary bounded linear transformation of \( {\\mathcal{L}}_{1}\\left( {X;\\mathcal{E}}\\right) \) into \( \\mathcal{E} \) satisfying (8). If \( \\sum \) is an integrable simple \( \\mathcal{E} \) -valued mapping on \( X \) and\n\n\[ \\sum = \\mathop{\\sum }\\limits_{{i = 1}}^{N... | Yes |
Let \( \mathcal{E} \) and \( \mathcal{F} \) be Banach spaces and let \( T \) be an element of \( \mathcal{L}\left( {\mathcal{E},\mathcal{F}}\right) \) . If an \( \mathcal{E} \) -valued mapping \( \Phi \) is integrable over an interval \( \left\lbrack {a, b}\right\rbrack \) with respect to an integrator \( \alpha \), th... | \[ T\left( {{\int }_{a}^{b}\Phi \left( t\right) {d\alpha }\left( t\right) }\right) = {\int }_{a}^{b}T\left( {\Phi \left( t\right) }\right) {d\alpha }\left( t\right) . \] | Yes |
Let \( \mathcal{E} \) be a Banach space, let \( \Phi \) be a norm-bounded \( \mathcal{E} \) -valued mapping defined on a real interval \( \left\lbrack {a, b}\right\rbrack \), and let \( \alpha \) be a bounded complex-valued function defined on \( \left\lbrack {a, b}\right\rbrack \) . If \( \Phi \) is integrable with re... | \[ {\int }_{a}^{b}\Phi \left( t\right) {d\alpha }\left( t\right) = \left( {{\int }_{a}^{c} + {\int }_{c}^{b}}\right) \Phi \left( t\right) {d\alpha }\left( t\right) \] | Yes |
Theorem 17.21. Let \( U \) be an open set of complex numbers, let \( \mathcal{E} \) be a Banach space, and let \( \Phi \) be a locally analytic \( \mathcal{E} \) -valued mapping defined on \( U \) (Ex. 15G). If \( \gamma \) is an arbitrary finite formal sum of closed rectifiable arcs in \( U \) such that \( \gamma \sim... | Proof. For any bounded linear functional fon \( \mathcal{E} \) the complex-valued function \( f \circ \Phi \) is locally analytic on \( U \), so \( {\int }_{\gamma }f\left( {\Phi \left( \zeta \right) }\right) {d\zeta } = 0 \) by the Cauchy-Goursat theorem for locally analytic scalar-valued functions. By Proposition 17.... | Yes |
Proposition 17.22. Let \( U \) be an open subset of \( \mathbb{C} \), let \( K \) be a compact subset of \( U \), and suppose given two oriented envelopes \( {\gamma }_{1} \) and \( {\gamma }_{2} \) of \( K \) in \( U \). Then \[ {\int }_{{\gamma }_{1}}\Phi \left( \zeta \right) {d\zeta } = {\int }_{{\gamma }_{2}}\Phi \... | Proof. The winding number of \( {\gamma }_{1} - {\gamma }_{2} \) on \( \mathbb{C} \smallsetminus U \) is zero since \( {w}_{{\gamma }_{1}} = {w}_{{\gamma }_{2}} = 0 \) there. The winding number of \( {\gamma }_{1} - {\gamma }_{2} \) on \( K \) is likewise \( {w}_{{\gamma }_{1}} - {w}_{{\gamma }_{2}} = 0 \). Hence \( {\... | Yes |
Proposition 17.23. Let \( x \) be an element of a unital Banach algebra \( \mathcal{A} \), and let \( f \) be a locally analytic function on an open neighborhood \( U \) of \( {\sigma }_{\mathcal{A}}\left( x\right) \) . Then \( f\left( x\right) \) doubly commutes with \( x \), that is, \( f\left( x\right) \) commutes w... | Proof. If \( \lambda \notin {\sigma }_{\mathcal{A}}\left( x\right) \) then it is clear that \( {R}_{x}\left( \lambda \right) \) doubly commutes with \( \lambda - x \) , and therefore with \( x \) . Hence if \( \gamma \) is an oriented envelope of \( {\sigma }_{\mathcal{A}}\left( x\right) \) in \( U \), and \( y \) is a... | Yes |
Theorem 17.25. Let \( U \) be an open subset of \( \mathbb{C} \), and let \( {\mathcal{F}}_{U} \) denote the unital algebra of all locally analytic complex-valued functions defined on \( U \) . If \( \mathcal{A} \) is a unital Banach algebra and \( x \) is an element of \( \mathcal{A} \) such that \( {\sigma }_{\mathca... | Proof. That \( \rho \) is an algebra homomorphism was established in Theorem 17.24. To see that \( \rho \) is bounded with respect to \( \sigma \), note that there exists an oriented envelope \( \gamma \) of \( {\sigma }_{\mathcal{A}}\left( x\right) \) in \( {K}^{ \circ } \) . If \( L \) denotes the length of \( \gamma... | "No" |
Proposition 17.26. Let \( \mathcal{A} \) be a unital Banach algebra, and let \( {x}_{0} \) be an element of \( \mathcal{A} \) such that \( {\sigma }_{\mathcal{A}}\left( {x}_{0}\right) \) is contained in an open set \( U \) in the complex plane. Suppose given a sequence \( \left\{ {f}_{n}\right\} \) of locally analytic ... | Proof. Let \( \gamma \) be a fixed oriented envelope of \( {\sigma }_{\mathcal{A}}\left( {x}_{0}\right) \) in \( U \), let \( {W}_{\gamma } \) denote the range of \( \gamma \), and let \( V = \left\{ {\lambda \in U : {w}_{\gamma }\left( \lambda \right) = 1}\right\} \) . Then \( V \) is an open neighborhood of \( {\sigm... | Yes |
Proposition 17.27. If \( \mathcal{A} \) is a unital Banach algebra and \( f \) is a locally analytic function on some open neighborhood \( U \) of the spectrum of an element \( x \) of \( \mathcal{A} \), then \( {\sigma }_{\mathcal{A}}\left( {f\left( x\right) }\right) = f\left( {{\sigma }_{\mathcal{A}}\left( x\right) }... | Proof. Since \( \left( {\alpha - f}\right) \left( x\right) = \alpha - f\left( x\right) \) it suffices to show that \( f\left( x\right) \) is invertible in \( \mathcal{A} \) when and only when the function \( f \) has no zero in \( {\sigma }_{\mathcal{A}}\left( x\right) \) . One way is easy enough. If \( f\left( \lambda... | Yes |
Proposition 17.28. Let \( x \) be an element of a unital Banach algebra \( \mathcal{A} \), let \( g \) be a locally analytic function defined on some open neighborhood \( U \) of \( {\sigma }_{\mathcal{A}}\left( x\right) \), and suppose given a second locally analytic function \( f \) defined on some open set \( \widet... | Proof. That the function \( f\left( {g\left( x\right) }\right) \) is defined follows from the preceding result. Moreover, the set \( \{ \lambda \in U : g\left( \lambda \right) \in \widetilde{U}\} \) is an open neighborhood of \( {\sigma }_{\mathcal{A}}\left( x\right) \) on\n\nwhich \( f \circ g \) is defined and locall... | Yes |
Theorem 18.3 (Riesz Representation Theorem). Let \( X \) be a compact Hausdorff space. Then for each functional \( \varphi \) in \( \mathcal{C}{\left( X\right) }^{ * } \) there is a unique, regular, complex Borel measure \( \xi \) in \( {\mathcal{M}}_{0}\left( X\right) \) such that \[ \varphi \left( f\right) = {\varphi... | Proof. It has already been seen (Prop. 10.16) that for any complex Borel measure \( \xi \) on \( X \), regular or not, the functional \( {\varphi }_{\xi } \) defined in (1) is linear and satisfies the inequality \( \begin{Vmatrix}{\varphi }_{\xi }\end{Vmatrix} \leq \left| \xi \right| \left( X\right) = \parallel \xi \pa... | Yes |
A self-conjugate subalgebra \( \mathcal{A} \) of \( \mathcal{C}\left( X\right) \) coincides with \( \mathcal{C}\left( X\right) \) if and only if the real algebra \( {\mathcal{A}}_{\mathbb{R}} \) of real-valued functions in \( \mathcal{A} \) coincides with the real algebra \( {\mathcal{C}}_{\mathbb{R}}\left( X\right) \)... | Proof. All three assertions are immediate consequences of the fact that a function \( f \) belongs to \( \mathcal{A} \) if and only if \( \operatorname{Re}f \) and \( \operatorname{Im}f \) do (see Example \( 2\mathrm{H} \) ). | Yes |
Proposition18.8 (Kakutani). Every closed subalgebra \( \mathcal{A} \) of the algebra \( {\mathcal{C}}_{\mathbb{R}}\left( X\right) \) is also a sublattice of \( {\mathcal{C}}_{\mathbb{R}}\left( X\right) \) . | Proof. According to Problem 1D it suffices to verify that if \( f \in \mathcal{A} \), then \( \left| f\right| \in \mathcal{A} \) also. To establish the latter assertion observe that if we set \( {q}_{n}\left( t\right) = \) \( {p}_{n}\left( {1 - t}\right) \), where \( \left\{ {{p}_{n}\left( t\right) }\right\} \) is a se... | Yes |
Lemma 18.9. Suppose that \( \mathcal{A} \) is a strongly separating subalgebra of \( {\mathcal{C}}_{\mathbb{R}}\left( X\right) \) and that \( {x}_{1} \) and \( {x}_{2} \) are distinct points of \( X \) . Then for any two real numbers \( {t}_{1} \) and \( {t}_{2} \) there is a function \( f \) in \( \mathcal{A} \) such ... | Proof. To verify this, let \( \mathcal{L} = \left\{ {\left( {f\left( {x}_{1}\right) ,\mathrm{f}\left( {x}_{2}\right) }\right) : f \in \mathcal{A}}\right\} \) . Then \( \mathcal{L} \) is a nontrivial subalgebra of \( {\mathbb{R}}^{2} \), and since \( \mathcal{A} \) is strongly separating, it follows from Problem \( \mat... | No |
If \( X \) is a compact Hausdorff space, and if \( \mathcal{F} \) is any strongly separating set of functions in \( \mathcal{C}\left( X\right) \), then the linear submanifold of \( \mathcal{C}\left( X\right) \) generated algebraically by the set of all finite products of elements of \( \mathcal{F} \) and their complex ... | In both cases the system described is a self-conjugate strongly separating subalgebra. | No |
Proposition 18.11. For every compact subset \( K \) of \( \mathbb{C},\mathcal{P}\left( K\right) \) and \( \mathcal{A}\left( K\right) \) are closed subalgebras of \( \mathcal{C}\left( K\right) \) such that \( \mathcal{P}\left( K\right) \subset \mathcal{A}\left( K\right) \) . | Proof. Since every polynomial function clearly belongs to \( \mathcal{A}\left( K\right) \), it suffices to prove that \( \mathcal{P}\left( K\right) \) is a subalgebra of \( \mathcal{C}\left( K\right) \) and that \( \mathcal{A}\left( K\right) \) is closed. To show the former, let \( f \) and \( g \) be functions in \( \... | Yes |
Corollary 18.12. If \( K \) is an arbitrary compact subset of \( \mathbb{C} \), a necessary and sufficient condition for \( \mathcal{P}\left( K\right) \) to coincide with \( \mathcal{C}\left( K\right) \) is that the function \( \bar{\lambda } \) belong to \( \mathcal{P}\left( K\right) \). Similarly, \( \mathcal{A}\left... | Proof. If \( \bar{\lambda } \) belongs to either \( \mathcal{P}\left( K\right) \) or \( \mathcal{A}\left( K\right) \), then all polynomials \( p\left( {\lambda ,\bar{\lambda }}\right) \) in \( \lambda \) and \( \bar{\lambda } \) do so too; cf. Example D. If \( {K}^{ \circ } \neq \varnothing \), then \( \bar{\lambda } \... | No |
Proposition 18.13. If \( K \) is a compact subset of \( \mathbb{C} \) then the algebra \( \mathcal{P}\left( K\right) \) coincides with the restriction to \( K \) of the subalgebra \( \mathcal{P}\left( \widehat{K}\right) \) of \( \mathcal{C}\left( \widehat{K}\right) \) . Thus the functions in \( \mathcal{P}\left( K\righ... | Proof. A sequence \( \left\{ {p}_{n}\right\} \) of polynomials converging uniformly on \( K \) automatically converges uniformly on the outer boundary of \( K \) . But then, by the maximum modulus principle (Ex. 5M), \( \left\{ {p}_{n}\right\} \) converges uniformly on \( \widetilde{K} \) , and the limit is therefore d... | No |
Corollary 18.15. For any compact subset \( K \) of \( \mathbb{C} \) the Banach algebra \( \mathcal{R}\left( K\right) \) contains every function in \( \mathcal{A}\left( K\right) \) that can be continued analytically onto some open neighborhood of \( K \) . | As a matter of fact, Corollary 18.15 can be proved much more simply and directly. Indeed, if \( f \) is locally analytic on an open set \( U \) containing \( K \), and if \( \gamma \) is an oriented envelope of \( K \) in \( U \), then any suitably chosen Riemann sum approximating the Cauchy integral \[ \frac{1}{2\pi i... | Yes |
Proposition 18.16 (Runge’s Theorem). Let \( U \) be an open subset of \( \mathbb{C} \) and let \( P \) be a subset of \( \widehat{\mathbb{C}} \smallsetminus U \) with the property that \( P \) contains at least one point of each connected component of \( \widehat{\mathbb{C}} \smallsetminus U \) . Then for any locally a... | Proof. If \( U = \mathbb{C} \) then \( P = \{ \infty \}, f \) is entire, and the partial sums of any power series expansion of \( f \) may be used. Otherwise, for each positive integer \( n \) let \( {D}_{n}^{ - } \) denote the closed disc \( \{ \lambda \in \mathbb{C} : \left| \lambda \right| \leq n\} \), and set\n\n\[... | Yes |
Proposition 19.3. If \( {\left\{ {x}_{\gamma }\right\} }_{\gamma \in \Gamma } \) and \( {\left\{ {y}_{\gamma }\right\} }_{\gamma \in \Gamma } \) are two summable families of vectors in a normed space \( \mathcal{E} \), both indexed by the same index set \( \Gamma \), and if\n\n\[ x = \mathop{\sum }\limits_{{\gamma \in ... | Proof. These facts follow immediately from the definition of summability. | No |
Proposition 19.4 (Cauchy Criterion). An indexed family \( {\left\{ {x}_{\gamma }\right\} }_{\gamma \in \Gamma } \) of vectors in a Banach space is summable if and only if for every \( \varepsilon > 0 \) there exists a finite set of indices \( {D}_{c} \) such that \( \begin{Vmatrix}{{s}_{D} = \mathop{\sum }\limits_{{\ga... | Proof. The proof of the necessity of the condition is trivial, as usual, and will be omitted. Suppose then that \( \mathcal{E} \) is a Banach space and that \( {\left\{ {x}_{\gamma }\right\} }_{\gamma \in \Gamma } \) is an indexed family in \( \mathcal{E} \) satisfying the stated condition. Let \( \varepsilon \) be a p... | No |
Corollary 19.5. If \( {\left\{ {x}_{\gamma }\right\} }_{\gamma \in \Gamma } \) is a summable family of vectors in a normed space \( \mathcal{E} \) , then \( \left\{ {\gamma \in \Gamma : {x}_{\gamma } \neq 0}\right\} \) is countable. | Proof. For each positive integer \( n \), let \( {D}_{n} \) be a finite set of indices such that \( \begin{Vmatrix}{s}_{D}\end{Vmatrix} < 1/n \) whenever \( D \cap {D}_{n} = \varnothing \), and let\n\n\[ J = \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{D}_{n} \]\n\nIf \( \gamma \notin J \), then \( \{ \gamma \} \cap {D... | Yes |
Corollary 19.6. Let \( \mathcal{E} \) be a Banach space and let \( {\left\{ {x}_{\gamma }\right\} }_{\gamma \in \Gamma } \) be an indexed family of vectors in &A sufficient condition for the family \( \left\{ {x}_{\gamma }\right\} \) to be summable is for it to be absolutely summable, i.e., for the family of norms \( \... | Proof. For any finite set of indices \( D \subset \Gamma \) we have\n\n\[ \begin{Vmatrix}{s}_{D}\end{Vmatrix} \leq \mathop{\sum }\limits_{{\gamma \in D}}\begin{Vmatrix}{x}_{\gamma }\end{Vmatrix} \]\n\nand the result follows by Proposition 19.4. | Yes |
Theorem 19.8. Let \( \mathcal{E} \) be a (separable) Banach space, let \( {\left\{ {x}_{n}\right\} }_{n = 0}^{\infty } \) be a basis for \( \mathcal{E} \), and for each nonnegative integer \( n \) and vector \( x \) in \( \mathcal{E} \) let us write \( {\alpha }_{n}\left( x\right) \) for the \( n \) th coefficient of \... | Proof. As has been noted, the uniqueness requirement in the definition of a basis ensures that \( \left\{ {x}_{n}\right\} \) is linearly independent, so Proposition 19.7 applies. Hence the linear space \( \mathcal{L} \) of all the coordinate sequences of vectors in \( \mathcal{E} \) (with respect to the given basis \( ... | Yes |
Lemma 19.9. Let \( \left( {X, A}\right) \) be a biorthogonal system for a Banach space & as in the foregoing definition. Then the expansion operators \( {E}_{N} \) of the system \( \left( {X, A}\right) \) have the property that \( {E}_{M}{E}_{N} = {E}_{N}{E}_{M} = {E}_{M \land N} \) for every pair of nonnegative intege... | Proof. These assertions are immediate consequences of the biorthogonality relations. | No |
Proposition 19.10. Let \( \mathcal{E} \) be a Banach space, let \( X = {\left\{ {x}_{n}\right\} }_{n = 0}^{\infty } \) and \( A = {\left\{ {\alpha }_{n}\right\} }_{n = 0}^{\infty } \) be sequences in \( \mathcal{E} \) and \( {\mathcal{E}}^{ * } \), respectively, such that \( \left( {X, A}\right) \) is a biorthogonal sy... | Proof. If \( X \) is a basis for \( \mathcal{M} \), then for each vector \( x \) in \( \mathcal{M} \) we have \( x = \) \( \mathop{\lim }\limits_{N}{E}_{N}x \), from which it follows at once (by the uniform boundedness theorem; Theorem 12.15) that the sequence \( \left\{ {E}_{N}\right\} \) is uniformly bounded on \( \m... | Yes |
Theorem 3 (Hensel’s lemma). Let \( F\left( x\right) = {c}_{0} + {c}_{1}x + \cdots + {c}_{n}{x}^{n} \) be a polynomial whose coefficients are p-adic integers. Let \( {F}^{\prime }\left( x\right) = {c}_{1} + 2{c}_{2}x + 3{c}_{3}{x}^{2} + \cdots + n{c}_{n}{x}^{n - 1} \) be the derivative of \( F\left( x\right) \) . Let \(... | Proof of Hensel's LEMMA. I claim that there exists a unique sequence of rational integers \( {a}_{1},{a}_{2},{a}_{3},\ldots \) such that for all \( n \geq 1 \) : (1) \( F\left( {a}_{n}\right) \equiv 0\left( {\;\operatorname{mod}\;{p}^{n + 1}}\right) \) . (2) \( {a}_{n} \equiv {a}_{n - 1}\left( {\;\operatorname{mod}\;{p... | Yes |
Theorem 4.\n\n\[ \zeta \left( {2k}\right) = {\left( -1\right) }^{k}{\pi }^{2k}\frac{{2}^{{2k} - 1}}{\left( {{2k} - 1}\right) !}\left( {-\frac{{B}_{2k}}{2k}}\right) . \] | Proof. First take the logarithm of both sides of\n\n\[ \sinh \left( {\pi x}\right) = {\pi x}\mathop{\prod }\limits_{{n = 1}}^{\infty }\left( {1 + \frac{{x}^{2}}{{n}^{2}}}\right) \]\n\n(for \( x > 0 \) ). On the left we get\n\n\[ \log \sinh \left( {\pi x}\right) = \log \left\lbrack {\left( {{e}^{\pi x} - {e}^{-{\pi x}}}... | Yes |
Theorem 5. Let \( {d}_{k} \) be the least common denominator of the coefficients of \( {B}_{k}\left( x\right) \) . Thus: \( {d}_{1} = 2,{d}_{2} = 6,{d}_{3} = 2 \), etc. Then \[ {d}_{k}{\mu }_{k,\alpha }\left( {a + \left( {p}^{N}\right) }\right) \equiv {d}_{k}k{a}^{k - 1}{\mu }_{1,\alpha }\left( {a + \left( {p}^{N}\righ... | Proof. By Exercise 1 below, the polynomial \( {B}_{k}\left( x\right) \) starts out \[ {B}_{0}{x}^{k} + k{B}_{1}{x}^{k - 1} + \cdots = {x}^{k} - \frac{k}{2}{x}^{k - 1} + \cdots . \] Now \[ {d}_{k}{\mu }_{k,\alpha }\left( {a + \left( {p}^{N}\right) }\right) = {d}_{k}{p}^{N\left( {k - 1}\right) }\left( {{B}_{k}\left( \fra... | Yes |
Theorem 6. Let \( \mu \) be a p-adic measure on \( X \), and let \( f : X \rightarrow {\mathbb{Q}}_{p} \) be a continuous function. Then the Riemann sums\n\n\[ \n{S}_{N,\left\{ {x}_{a, N}\right\} }\underset{\text{ def }}{ = }\mathop{\sum }\limits_{\substack{{0 \leq a < {p}^{N}} \\ {a + \left( {p}^{N}\right) \subset X} ... | Proof. Suppose that \( {\left| \mu \left( U\right) \right| }_{p} \leq B \) for all compact-open \( U \subset X \) . We first estimate for \( M > N \)\n\n\[ \n{\left| {S}_{N,\left\{ {x}_{a, N}\right\} } - {S}_{M,\left\{ {x}_{a, M}\right\} }\right| }_{p}\n\]\n\nBy writing \( X \) as a finite union of intervals, we can ch... | Yes |
(1) If \( p - 1 \nmid k \), then \( {\left| {B}_{k}/k\right| }_{p} \leq 1 \) . | To prove (1), we write (assuming \( k > 1 \) ; if \( k = 1 \) and \( p > 2 \), then \( \left. {{\left| {B}_{1}/1\right| }_{p} = {\left| -1/2\right| }_{p} = 1}\right) \) : \n\n\[ \n{\left| {B}_{k}/k\right| }_{p} = {\left. {\left. {\left. \left| 1/\left( {\alpha }^{-k} - 1\right) \right| \right| }_{p}\left| 1/\left( 1 - ... | Yes |
Theorem 8. For fixed \( p \) and fixed \( {s}_{0},{\zeta }_{p,{s}_{0}}\left( s\right) \) is a continuous function of \( s \) which does not depend on the choice of \( \alpha \in \mathbb{Z}, p \nmid \alpha ,\alpha \neq 1 \), which appears in its definition. | Proof. It is clear that \( §2 \) and the corollary at the end of \( §5 \) imply that the integral is a continuous function of \( s \) . The factor \( 1/\left( {{\alpha }^{-\left( {{s}_{0} + \left( {p - 1}\right) s}\right) } - 1}\right) \) is a continuous function as long as we don’t allow \( s = 0 \) when \( {s}_{0} = ... | Yes |
Theorem 9. Let \( F \) be a finite field containing \( q \) elements, and let \( f = \left\lbrack {F : {\mathbb{F}}_{p}}\right\rbrack \) (i.e., the dimension of \( F \) as a vector space over its prime field \( {\mathbb{F}}_{p} \) ). Let \( K \) be an algebraic closure of \( {\mathbb{F}}_{p} \) containing \( F \) . The... | Proof. Since \( F \) is an \( f \) -dimensional vector space over \( {\mathbb{F}}_{p} \), the number of elements is equal to the number of choices of the \( f \) components (i.e.,\ | No |
Theorem 10. If \( V \) is a finite dimensional vector space over a locally compact field \( F \), then all norms on \( V \) are equivalent. | Proof. Let \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) be a basis for \( V \) . Define the sup-norm \( \parallel {\parallel }_{\text{sup }} \) (pronounced \ | No |
Theorem 11. Let \( K \) be a finite extension of \( {\mathbb{Q}}_{p} \). Then there exists a field norm on \( K \) which extends the norm \( {\left| \right| }_{p} \) on \( {\mathbb{Q}}_{p} \). | Proof. Let \( n = \left\lbrack {K : {\mathbb{Q}}_{p}}\right\rbrack \). We first define \( {\left| \right| }_{p} \) on \( K \), and then prove that it’s really a field norm on \( {\left. K\text{extending} \mid \right| }_{p} \) on \( {\mathbb{Q}}_{p} \). For any \( \alpha \in K \) we define\n\n\[{\left| \alpha \right| }_... | Yes |
Theorem 13. \( \Omega \) is algebraically closed. | Proof. Let: \( f\left( X\right) = {X}^{n} + {a}_{n - 1}{X}^{n - 1} + \cdots + {a}_{1}X + {a}_{0},{a}_{i} \in \Omega \) . We must show that \( f\left( X\right) \) has a root in \( \Omega \) . For each \( i = 0,1,\ldots, n - 1 \), let \( {\left\{ {a}_{ij}\right\} }_{j} \) be a sequence of elements of \( {Q}_{p} \) which ... | No |
Lemma 1. Every \( f\left( X\right) \in {\mathbb{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) converges in \( D\left( {1}^{ - }\right) \) . | Proof. Let \( f\left( X\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{X}^{n},{a}_{n} \in {\mathbb{Z}}_{p} \), and let \( x \in D\left( {1}^{ - }\right) \) . Thus, \( {\left| x\right| }_{p} < 1 \) . Also \( {\left| {a}_{n}\right| }_{p} \leq 1 \) for all \( n \) . Hence \( {\left| {a}_{n}{x}^{n}\right| }_{p} ... | Yes |
Lemma 2. Every \( f\left( X\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{X}^{n} \in \Omega \left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) which converges in an (open or closed) disc \( D = D\left( r\right) \) or \( D\left( {r}^{ - }\right) \) is continuous on \( D \) . | Proof. Suppose \( {\left| {x}^{\prime } - x\right| }_{p} < \delta \), where \( \delta < {\left| x\right| }_{p} \) will be chosen later. Then \( {\left| {x}^{\prime }\right| }_{p} = {\left| x\right| }_{p} \) . (We are assuming \( x \neq 0 \) ; the case \( x = 0 \) is very easy to check separately.) We have\n\n\[ \n{\lef... | Yes |
Lemma 3. Let \( F\left( X\right) = \sum {a}_{i}{X}^{i} \in 1 + X{\mathbb{Q}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) . Then \( F\left( X\right) \in 1 + X{\mathbb{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) if and only if \( F\left( {X}^{p}\right) /{\left( F\left( X\right) \righ... | Proof. If \( F\left( X\right) \in 1 + X{\mathbb{Z}}_{p}\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \), then, since \( {\left( a + b\right) }^{p} \equiv {a}^{p} + {b}^{p}\left( {\;\operatorname{mod}\;p}\right) \) and \( {a}^{p} \equiv a\left( {\;\operatorname{mod}\;p}\right) \) for \( a \in {\mathbb{Z}}_{p} \... | Yes |
Lemma 4. In the above notation, let \( f\left( X\right) = \left( {1 - X/{\alpha }_{1}}\right) \cdots \left( {1 - X/{\alpha }_{n}}\right) \) be the factorization of \( f\left( X\right) \) in terms of its roots \( {\alpha }_{i} \in \Omega \) . Let \( {\lambda }_{i} = {\operatorname{ord}}_{p}1/{\alpha }_{i} \) . Then, if ... | Proof. We may suppose the \( {\alpha }_{i} \) to be arranged so that \( {\lambda }_{1} \leq {\lambda }_{2} \leq \cdots \leq {\lambda }_{n} \) . Say \( {\lambda }_{1} = {\lambda }_{2} = \cdots = {\lambda }_{r} < {\lambda }_{r + 1} \) . We first claim that the first segment of the Newton polygon is the segment joining \(... | Yes |
Lemma 5. Let \( b \) be the least upper bound of all slopes of the Newton polygon of \( f\left( X\right) = 1 + \mathop{\sum }\limits_{{i = 1}}^{\infty }{a}_{i}{X}^{i} \in 1 + {X\Omega }\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) . Then the radius of convergence is \( {p}^{b} \) (b may be infinite, in whic... | Proof. First let \( {\left| x\right| }_{p} < {p}^{b} \), i.e., \( {\operatorname{ord}}_{p}x > - b \) . Say \( {\operatorname{ord}}_{p}x = - {b}^{\prime } \), where \( {b}^{\prime } < b \) . Then \( {\operatorname{ord}}_{p}\left( {{a}_{i}{x}^{i}}\right) = {\operatorname{ord}}_{p}{a}_{i} - i{b}^{\prime } \) . But it is c... | Yes |
Lemma 8. Let \( f\left( X\right) = 1 + \mathop{\sum }\limits_{{i = 1}}^{\infty }{a}_{i}{X}^{i} \in 1 + {X\Omega }\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) converge and have value 0 at \( \alpha \) . Let \( g\left( X\right) = 1 + \mathop{\sum }\limits_{{i = 1}}^{\infty }{b}_{i}{X}^{i} \) be obtained by d... | Proof. Let \( {f}_{n}\left( X\right) = 1 + \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{X}^{i} \) . Clearly,\n\n\[ \n{b}_{i} = 1/{\alpha }^{i} + {a}_{1}/{\alpha }^{i - 1} + {a}_{2}/{\alpha }^{i - 2} + \cdots + {a}_{i - 1}/\alpha + {a}_{i},\n\]\n\nso that\n\n\[ \n{b}_{i}{\alpha }^{i} = {f}_{i}\left( \alpha \right)\n\]\n\n... | Yes |
Theorem 14 ( \( p \) -adic Weierstrass Preparation Theorem). Let \( f\left( X\right) = 1 + \) \( \mathop{\sum }\limits_{{i = 1}}^{\infty }{a}_{i}{X}^{i} \in 1 + {X\Omega }\left\lbrack \left\lbrack X\right\rbrack \right\rbrack \) converge on \( D\left( {p}^{\lambda }\right) \) . Let \( N \) be the total horizontal lengt... | Proof. We use induction on \( N \) . First suppose \( N = 0 \) . Then we must show that \( g\left( X\right) \), the inverse power series of \( f\left( X\right) \), converges and is nonzero on \( D\left( {p}^{\lambda }\right) \) . This was part of Exercise 3 of \( \$ \) IV.1, but, since this is an important fact, we'll ... | No |
Lemma 1. \( Z\left( {{H}_{f}/{\mathbb{F}}_{q};T}\right) \) has coefficients in \( \mathbb{Z} \) . | Proof. We consider the \( K \) -points \( P = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) of \( {H}_{f}(K \) a finite extension of \( {\mathbb{F}}_{q} \) ) according to the least \( s = {s}_{0} \) for which all \( {x}_{i} \in {\mathbb{F}}_{{q}^{{s}_{0}}} \) . If \( {P}_{j} = \left( {{x}_{1j},\ldots ,{x}_{nj}}\right) \) ... | No |
Lemma 2. The coefficient of \( {T}^{j} \) in \( Z\left( {{H}_{f}/{\mathbb{F}}_{q};T}\right) \) is \( \leq {q}^{nj} \) . | Proof. The maximum value for \( {N}_{s} \) is \( {q}^{ns} = \# {\mathbb{A}}_{{\mathbb{F}}_{{q}^{s}}}^{n} \) . The coefficients of \( Z\left( {{H}_{f}/{\mathbb{F}}_{q};T}\right) \) are clearly less than or equal to the coefficients of the series with \( {N}_{s} \) replaced by \( {q}^{ns} \) . But\n\n\[ \exp \left( {\mat... | Yes |
Lemma 3. Let \( G \in {R}_{0} \), and let \( \Psi = {\Psi }_{q, G} \). Then \( \operatorname{Tr}\left( {\Psi }^{s}\right) \) converges for \( s = 1,2,3,\ldots \), and\n\n\[ \n{\left( {q}^{s} - 1\right) }^{n}\operatorname{Tr}\left( {\Psi }^{s}\right) = \mathop{\sum }\limits_{\substack{{x \in {\Omega }^{n}} \\ {{x}^{{q}^... | Proof. We first prove the lemma for \( s = 1 \), and then easily reduce the general case to this special case. Since \( \Psi \left( {X}^{u}\right) = \mathop{\sum }\limits_{{v \in U}}{g}_{{qv} - u}{X}^{v} \), we have\n\n\[ \n\operatorname{Tr}\Psi = \mathop{\sum }\limits_{{u \in U}}{g}_{\left( {q - 1}\right) u}\n\]\n\nwh... | Yes |
Lemma 4. If \( G\left( X\right) = \mathop{\sum }\limits_{{w \in U}}{g}_{w}{X}^{w} \in {R}_{0} \) and \( \Psi = {T}_{q} \circ G \), so that \( \Psi \) has matrix \( A = {\left\{ {g}_{{qv} - u}\right\} }_{v, u \in U} \), then the series \( \det \left( {1 - {AT}}\right) \) is a well-defined element of \( \Omega \left\lbra... | \[ {\exp }_{p}\left\{ {-\mathop{\sum }\limits_{{s = 1}}^{\infty }\operatorname{Tr}\left( {A}^{s}\right) {T}^{s}/s}\right\} \] | Yes |
Lemma 5. Let \( F\left( T\right) = \mathop{\sum }\limits_{{i = 0}}^{\infty }{a}_{i}{T}^{i} \in K\left\lbrack \left\lbrack T\right\rbrack \right\rbrack \), where \( K \) is any field. For \( m, s \geq 0 \) , let \( {A}_{s, m} \) be the matrix \( {\left\{ {a}_{s + i + j}\right\} }_{0 \leq i, j \leq m} \) :\n\n\[ \left( \... | Proof. First suppose that \( F\left( T\right) \) is such a quotient. Let \( P\left( T\right) = \mathop{\sum }\limits_{{i = 0}}^{M}{b}_{i}{T}^{i} \) , \( Q\left( T\right) = \mathop{\sum }\limits_{{i = 0}}^{N}{c}_{i}{T}^{i} \) . Then, since \( F\left( T\right) \cdot Q\left( T\right) = P\left( T\right) \), equating coeffi... | Yes |
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