Life without "Choice"
We propose an AC-independent proof of the existence of a non-measurable set as a consequence of the Hahn-Banach theorem of functional analysis which is known to be strictly weaker than AC.
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arXiv subjects
M. A. Sofi
Publications and source records attributed to M. A. Sofi.
Implications of an affirmative solution to the Lindenstrauss Problem
The question regarding the location of Banach spaces inside their biduals has been investigated and answered reasonably satisfactorily in the linear theory of Banach spaces. Thus, for instance, whereas it is known that a dual Banach space is complemented inside its bidual, the space of all null sequences is not! However, the latter space is a Lipschitz retract of its bidual. In his famous paper of 1964, Lindenstrauss asked if every Banach space is a Lipschitz retract of its bidual. In this short note, we show how to relate the Lindenstrauss problem (LP) to certain other important and well-known questions that remain open in the Lipschitz theory of Banach spaces and how these latter questions may be settled in the affirmative under the assumption of (LP) having a positive solution.
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On some new metric characterisations of Hilbert spaces
In the literature surrounding the theory of Banach spaces, considerable effort has been invested in exploring the conditions on a Banach space X that characterise X as being an inner product space or as a linearly isomorphic copy of a Hilbert space. On the other hand, a different theory emerges when the class of Banach spaces is looked upon as a Lipschitz category where Lipschitz maps are used as morphisms in the new category in place of the familiar bounded linear maps in the linear theory. This paper provides a short survey of recent results involving the appropriate Lipschitz analogues of certain well known results from the linear theory characterizing Hilbert spaces. Whereas isometric description of Hilbert spaces has all along been a popular theme in this line of investigations, we shall concentrate mainly on isomorphic characterisations which entail the existence of an equivalent norm on the underlying space arising from an inner product.
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Nonlinear retracts and the geometry of Banach spaces
In the nonlinear geometry of Banach spaces where the objects in the category are Banach spaces as in the linear case, the morphisms in the new setting are taken to comprise of certain nonlinear maps involving say, Lipschitz maps and, in some cases, uniformly continuous, coarse or coarse Lipschitz mappings arising from the underlying metric or the uniform structure attached to the norm of the Banach space. The question as to what extent the Lipschitz or the uniform structure may be used to capture the full linear structure of a Banach space has been one of the most fundamental problems pertaining to the nonlinear structure of Banach spaces since this line of investigation was undertaken by Lindenstrauss in the late sixties. This line of research which is subsumed under the so called Ribe program broadly underscores the view that metric spaces encode a much deeper and hidden structure than is apparent. It is truly surprising how the linear structure of a Banach space gets captured to a considerable extent by its metric space structure. This point of view has led to deep insights into Banach space theory that has paved the way for these ideas being employed in seemingly unrelated disciplines including harmonic analysis, geometric group theory and many other domains of mathematics.
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Extension operators and nonlinear structure of Banach spaces
The problem involving the extension of functions from a certain class and defined on subdomains of the ambient space to the whole space is an old and a well investigated theme in analysis. A related question whether the extensions that result in the process may be chosen in a linear or a continuous manner between appropriate spaces of functions turns out to be highly nontrivial. That this holds for the class of continuous functions defined on metric spaces is the well-known Borsuk-Dugundji theorem which asserts that given a metric space M and a subspace S of M, each continuous function g on S can be extended to a continuous function f on X such that the resulting assignment from C(S) to C(M) is a norm-one continuous linear extension operator. The present paper is devoted to an investigation of this problem in the context of extendability of Lipschitz functions from closed subspaces of a given Banach space to the whole space such that the choice of the extended function gives rise to a bounded linear (extension) operator between appropriate spaces of Lipschitz functions. It is shown that the indicated property holds precisely when the underlying space is isomorphic to a Hilbert space. Among certain useful consequences of this theorem, we provide an isomorphic analogue of a well-known theorem of S. Reich by show ing that closed convex subsets of a Banach space X arise as Lipschitz retracts of X precisely when X is isomorphically a Hilbert space. We shall also discuss the issue of bounded linear extension operators between spaces of Lipschitz functions now defined on arbitrary subsets of Banach spaces and provide a direct proof of the known non-existence of such an extension operator by using methods which are more accessible than those initially employed by the authors.
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Embeddability, representability and universality involving Banach spaces
Given a category of objects, it is both useful and important to know if all the objects in the category may be realised as sub-objects -- via morphisms in the given category -- of a single object in that category enjoying some nice properties. In the category of separable Banach spaces with morphisms consisting of linear isometries, such an example of (a universal) object is provided by the well-known Banach Mazur theorem: the space C[0,1] of continuous functions on the unit interval contains each separable Banach spaces as a closed subspace via a linear isometry. Here the question also arises if, as opposed to realising (separable) Banach spaces as spaces of continuous functions on [0, 1], it is possible to embed a Banach space as a subgroup of the group of linear isometries (resp. unitaries) on a nice Banach (resp. Hilbert) space. If such is the case, one says that the given Banach space is representable as a group of isometries (resp. unitaries). On the other hand, the idea of embeddability involves the possibility of realising each object in a given class of objects as included inside another object of the same class enjoying some good properties which are not present in the initial object. Further, considering that a Banach space also comes equipped with weaker structures involving the underlying metric (Lipschitz), uniform and topological structures, it follows that besides the linear isomorphisms (isometries), one may also consider morphisms in this category consisting of maps which are Lipschitz, uniformly continuous or continuous. This motivates the consideration of situations where it becomes necessary to know if a Banach space (resp a metric space) may be embedded in a nice Banach space as a metric, uniform or merely as a topological space.
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Banach limits -- Some new thoughts and perspectives
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. Some open problems that naturally arise in the study have also been included.
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Riemann integrability under weaker forms of continuity in infinite dimensional spaces
In classical analysis, the relationship between continuity and Riemann integrability is an intimate one: a continuous function on a closed and bounded interval is always Riemann integrable whereas a Riemann integrable function is continuous almost everywhere. In the setting of functions taking values in infinite dimensional spaces that include quasi Banach spaces, one encounters certain curious situations 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 some of 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 ( generally compatible) linear topologies on X (resp. its dual) 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 Frechet spaces on the one hand and of quasi Banach spaces on the other. The paper concludes with a brief (but non-technical) description on recent developments and the current of art involving various other aspects of vector integration
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Some problems in functional analysis inspired by Hahn Banach type theorems
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.
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