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Zinovij G. Sheftel - One of the best experts on this subject based on the ideXlab platform.
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Linear Continuous Functionals and Dual Spaces
1996Co-Authors: Yuri M. Berezansky, Zinovij G. SheftelAbstract:In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach Theorem on extensions of linear functionals and the Banach-Steinhaus Theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated.
Thurlow A Cook - One of the best experts on this subject based on the ideXlab platform.
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let e be a locally convex hausdorff linear topological vector space henceforth called simply a locally convex space a schauder basis for e is a sequence of vectors xn in e together with a sequence fn of continuous linear functionals from e such that fk xj bk j and for each xee x enc lfn x xn convergence being in the topology of e
2016Co-Authors: Thurlow A CookAbstract:For each positive integer m let Sm(x) = 1f.(x)xn. We call the basis (xn; fn) equicontinuous when the partial sum operators (Sm) are equicontinuous in the given topology of E. In this note a necessary and sufficient condition for a Schauder basis to be weakly equicontinuous is given and complete locally convex spaces with weakly equicontinuous Schauder bases are characterized. A form of the latter result has been obtained in [1, p. 268]. Equicontinuous bases were introduced in [3, p. 208]. If the space E is barrelled then by the Banach-Steinhaus Theorem, any Schauder basis of E is equicontinuous. However, if E is examined in its weak topology considerably different results occur. 1. Theorem. Let E be a locally convex space with a Schauder basis (xn; fn). Then this basis is weakly equicontinuous if and only if ffn } is a Hamnel basis for E*. Moreover, if E is complete, these conditions hold if and only if E is linearly homeomorphic to (s), the countable product of reals. PROOF. Suppose (Xn; fn) is weakly equicontinuous. It follows that
David Carfì - One of the best experts on this subject based on the ideXlab platform.
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The pointwise Hellmann-Feynman Theorem
2010Co-Authors: David CarfìAbstract:In this paper we study from a topological point of view the Hellmann-Feynman Theorem of Quantum Mechanics. The goal of the paper is twofold: On one hand we emphasize the role of the strong topology in the classic version of the Theorem in Hilbert spaces, for what concerns the kind of convergence required on the space of continuous linear endomorphisms, which contains the space of (continuous) observables. On the other hand we state and prove a new pointwise version of the classic Hellmann-Feynman Theorem. This new version is not yet present in the literature and follows the idea of A. Bohm concerning the topology which is desiderable to use in Quantum Mechanics. It is indeed out of question that this non-trivial new version of the Hellmann-Feynman Theorem is the ideal one - for what concerns the continuous observables on Hilbert spaces, both from a theoretical point of view, since it is the strongest version obtainable in this context - we recall that the pointwise topology is the coarsest one compatible with the linear structure of the space of continuous observables -, and from a practical point of view, because the pointwise topology is the easiest to use among topologies: it brings back the problems to the Hilbert space topology. Moreover, we desire to remark that this basic Theorem of Quantum Mechanics, in his most desiderable form, is deeply interlaced with two cornerstones of Functional Analysis: the Banach-Steinhaus Theorem and the Baire Theorem .
Yuri M. Berezansky - One of the best experts on this subject based on the ideXlab platform.
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Linear Continuous Functionals and Dual Spaces
1996Co-Authors: Yuri M. Berezansky, Zinovij G. SheftelAbstract:In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach Theorem on extensions of linear functionals and the Banach-Steinhaus Theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated.
Hajime Ishihara - One of the best experts on this subject based on the ideXlab platform.
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sequentially continuity in constructive mathematics
2001Co-Authors: Hajime IshiharaAbstract:The classical validity of many important Theorems of functional analysis, such as the Banach-Steinhaus Theorem, the open mapping Theorem and the closed graph Theorem, depends on Baire’s Theorem about complete metric spaces, which is an indispensable tool in this area. A form of Baire’s Theorem has a constructive proof [5, Theorem 1.3], but its classical equivalent, if a complete metric space is the union of a sequence of its subsets, then the closure of at least one set in the sequence must have nonempty interior which is used in the standard argument to prove that the above Theorems have no known constructive proof. If we could prove the Baire’s Theorem of the above form, we would have the following forms of constructive versions of Banach’s inverse mapping Theorem, the open mapping Theorem, the closed graph Theorem, the Banach-Steinhaus Theorem and the Hellinger-Toeplits Theorem: Theorem 1 (Banach’s inverse mapping Theorem) LetT be a one-one continuous linear mapping of a separable Banach space E onto a Banach space F. Then T-1 is continuous. Theorem 2 (The open mapping Theorem) Let T be a continuous linear mapping of a Banach space E onto a Banach space F such that ker(T) is located1. Then T is open. Theorem 3 (The closed graph Theorem) Let T be a linear mapping of a Banach space E into a Banach space F such that graph(T) is closed and separable. Then T is continuous. Theorem 4 (The Banach-Steinhaus Theorem) Let {Tm} be a sequence of continuous linear mappings from a separable Banach space E into a normed space F such that $$ Tx: = \mathop {\lim }\limits_{m \to \infty } {T_m}x $$ exists for all x ∈ E. Then T is continuous. Theorem 5 (The Hellinger-Toeplitz Theorem) Let T be a linear mapping from a Banach space E into a separable normed space with the following property: if f is a normable2 linear functional f on F, and {xn} converges to 0 in E, then f (Txn) → 0. Then T is continuous.
