Around finite-dimensionality in functional analysis: a personal perspective

Sofi, M. A.

doi:10.1007/s13398-012-0083-5

Cited by 4 publications

(5 citation statements)

References 25 publications

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“…It follows from Theorem 20 and the discussion preceding it that the property involving the existence of a Lipschitz selector H(X) → X is a finite dimensional (FD) property in the class of Banach spaces: a property (P) is said to be a (FD) property if it holds for finite dimensional Banach spaces but fails in each infinite dimensional Banch space. A comprehensive study of this phenomenon was undertaken in [16] where it was shown with the help of a host of examples that suitably formulated analogues of (FD) properties in the framework of Fréchet spaces lead to new insights into the structure of the latter class of spaces. It is interesting to note that in a fairly large number of cases, these (FD) properties hold in this new setting exactly if the underlying Fréchet space is nuclear!…”

Section: E Banach Limits and The Structure Of Banach Spacesmentioning

confidence: 99%

Banach limits: some new thoughts and perspectives

Sofi

2019

J Anal

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The existence of a Banach limit as a translation invariant positive continuous linear functional on the space of bounded scalar sequences which is equal to 1 at the constant sequence (1, 1, . . . , 1, . . .) is proved in a first course on functional analysis as a consequence of the Hahn Banach extension theorem. Whereas its use as an important tool in classical summability theory together with its application in the existence of certain invariant measures on compact (metric) spaces is well known, a renewed interest in the theory of Banach limits has led to certain applications which have opened new vistas in the structure of Banach spaces.The paper is devoted to a discussion of certain developments, both classical and recent, surrounding the theory of Banach limits including the structure of the set of Banach limits with special emphasis on certain aspects of their applications to the existence of certain invariant measures, vector valued analogues of Banach limits, functional equations and in the structure theory of Banach spaces involving the existence of selectors of certain multi-valued mappings into the metric space of non-empty, convex, closed and bounded subsets of a Banach space with respect to the Hausdorff metric. The paper shall conclude with a brief description of some recent results of the author on the study of 'simultaneous continuous linear' operators (linear selections) involving Hahn Banach extensions on spaces of Lipschitz functions on (subspaces of) Banach spaces.1.1. Existence of Banach limits. One of the common methods to prove the existence of Banach limits is via the use of the invariant version of the Hahn Banach theorem whereas the first proof by M.M. 0

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Section: E Banach Limits and The Structure Of Banach Spacesmentioning

confidence: 99%

Banach limits: some new thoughts and perspectives

Sofi

2019

J Anal

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“…In view of the role played by (FD)-properties in the context of Riemann integrability, it will be useful to spend some time on the theme of finite dimensionality in an infinite dimensional context (See [19] for a detailed treatment of this phenomenon). To this end, let us note that an important (FD)-property is provided by considering the socalled Hilbert-Schmidt property of a Banach space.…”

Section: Definition 35 ([19]mentioning

confidence: 99%

“…Example 3 (i) ( [19]) M (X)\M bv (X) together with the zero element contains an infinite dimensional space. This set is even known to be non-meagre.…”

Section: (A) Size Of the Setmentioning

confidence: 99%

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Riemann integrability under weaker forms of continuity in infinite dimensional spaces

Sofi

2016

Preprint

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In classical analysis, the relationship between continuity and Riemann integrability of a function is an intimate one: a continuous function f : [a, b] → R is always Riemann integrable whereas a Riemann integrable function is continuous almost everywhere (a.e). In the setting of functions taking values in infinite dimensional abstract spaces that include quasi Banach spaces, one encounters certain curious phenomena involving the breakdown of the above stated phenomena, besides the failure of the fundamental theorem of calculus and the non-existence of primitives for continuous functions! While these properties can surely be salvaged within the class of Banach spaces, it turns out that certain important properties involving vector integration that include Riemann integration no longer hold in an infinite dimensional setting. This will be seen to be the case, for example, in situations when it is required to integrate functions which are continuous with respect to certain well known linear topologies on X (resp. on X * ) weaker than the norm topology. As we shall see in Section 3(a), such a requirement imposes rather severe restrictions on the space in question. The present paper is devoted to a discussion of these issues which will be examined in the setting of Banach and Fréchet spaces on the one hand and of quasi Banach spaces on the other.

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“…In other words, ( * ) is a finite-dimensional-property in the following sense. For a detailed account of the material included in this section, see [38]. Definition 4.2.…”

Section: Hb-extension Property As An (Fd)-propertymentioning

confidence: 99%

Some problems in functional analysis inspired by Hahn-Banach type theorems

Sofi¹

2014

Ann. Funct. Anal.

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Abstract. As a cornerstone of functional analysis, Hahn-Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications of the Hahn-Banach theorem which are less familiar to the mathematical community, apart from highlighting certain aspects of the Hahn-Banach phenomena which have spurred intense research activity over the past few years, especially involving operator analogues and nonlinear variants of this theorem.For a discussion of a whole lot of issues related to the Hahn-Banach theorem not treated in this paper, the best source is a famous survey paper by Narici and Beckenstein [31] which deals, among other things, with the different settings witnessing the validity of the Hahn-Banach theorem.

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Around finite-dimensionality in functional analysis: a personal perspective

Cited by 4 publications

References 25 publications

Banach limits: some new thoughts and perspectives

Banach limits: some new thoughts and perspectives

Riemann integrability under weaker forms of continuity in infinite dimensional spaces

Some problems in functional analysis inspired by Hahn-Banach type theorems

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