Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)


Rao, T. S. S. R. K. (TSMU-Bangalore; ISI)

Abstract (Summary of the Work)

In this chapter, we explain the background and the main theme of this thesis and provide a chapter-wise summary of its principal results. We introduce some notations and preliminaries that will be used in the subsequent chapters.Study of proximinality related properties and ball intersection related properties of Banach spaces have been an active area of research in the field of geometry of Banach spaces. In this thesis, we mainly study these two classes of Banach space theoretic properties.We consider only Banach spaces over the real field R and all subspaces we consider are assumed to be closed.1.1 PreliminariesFor a Banach space X and a subspace Y , one of the basic problems in the field of approximation theory is the existence of a best approximation from Y for an element x of X. If this happens for every x ∈ X, then Y is said to be a proximinal subspace of X.Definition 1.1.1. Let K be a non-empty closed subset of a Banach space X. For x ∈ X, the distance of x from K, denoted by d(x, K), is given by d(x, K) = inf{kx − kk : k ∈ K}.The set-valued mapping PK : X → 2 K defined by PK(x) = {k ∈ K : d(x, K) = kx − kk} is called the metric projection onto K. An element of PK(x) is called a best approximation from K to x. The set K is said to be proximinal in X if PK(x) 6= ∅ for all x ∈ X.Some of the natural examples of proximinal subspaces are reflexive subspaces and ker(f), where f ∈ X∗ is such that kfk = f(x) for some x ∈ X with kxk = 1 (the so-called norm attaining functional). The earliest results concerning characterization of proximinal subspaces of finite co-dimension (that is dim(X/Y ) < ∞) are mainly due to Garkavi (see [17,18] for details). For instance, in [18], he characterized finite co-dimensional proximinal subspaces of C(K), the space of all continuous functions on a compact Hausdorff space K, equipped with the supremum norm.Theorem 1.1.2 ([18]). Let K be a compact Hausdorff space and let Y be a finite codimensional subspace of C(K). Then Y is proximinal in C(K) if and only if the annihilator Y ⊥ satisfies the following three conditions:(a) supp(µ +) T supp(µ −) = ∅ for each µ ∈ Y ⊥ \\ {0},(b) µ is absolutely continuous with respect to ν on supp(ν) for every pair µ, ν ∈ Y ⊥ \\{0},(c) supp(ν) \\ supp(µ) is closed for each pair µ, ν ∈ Y ⊥ \\ {0}.The following result by Garkavi gives a necessary condition for factor reflexive subspaces to be proximinal. We recall that a subspace Y of a Banach space X is said to be a factor reflexive subspace if the quotient space X/Y is reflexive.Proposition 1.1.3 ([46, Chapter III, Lemma 1.1]). If Y is a factor reflexive proximinal subspace of a Banach space X, then every f ∈ Y ⊥ is a norm attaining functional on X.But in general the converse of the Proposition 1.1.3 need not be true though it is true when Y is of co-dimension one. Precisely, for a Banach space X and f ∈ X∗ , ker(f) is proximinal in X if and only if f is a norm attaining functional on X.In [19], Godefroy and Indumathi introduced a stronger version of proximinality called strong proximinality.Definition 1.1.4. A proximinal subspace Y of a Banach space X is said to be strongly proximinal in X if for every x ∈ X and every ε > 0, there exists a δ > 0 such that PY (x, δ) ⊆ PY (x) + εBX, where PY (x, δ) = {y ∈ Y : kx − yk < d(x, Y ) + δ} and BX is the closed unit ball of X.For a proximinal subspace Y of a Banach space X, one can easily observe that the above definition is equivalent to the following: for every element x ∈ X and for every sequence (yn) in Y with kx − ynk → d(x, Y ), d(yn, PY (x)) → 0.Clearly, any finite dimensional subspace of a Banach space is strongly proximinal. In [38], Narayana proved that every infinite dimensional Banach space can be embedded isometrically as a non-strongly proximinal hyperplane in another Banach space.In [16], Franchetti and Pay´a introduced a non-smooth extension of Fr´echet differentiability, namely strong subdifferentiability, which in turn characterizes strongly proximinal hyperplanes.


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