Papers by Pradipta Bandyopadhyay
Archiv der Mathematik, 2021
In this short note, we answer two questions about Gurariy spaces asked in the literature in the a... more In this short note, we answer two questions about Gurariy spaces asked in the literature in the affirmative. We also prove the analogue of one of the results for the noncommutative Guariy space.
K. S. Lau had shown that a reflexive Banach space has the Mazur Intersection Property (MIP) if an... more K. S. Lau had shown that a reflexive Banach space has the Mazur Intersection Property (MIP) if and only if every closed bounded convex set is the closed convex hull of its farthest points. In this work, we show that in general this latter property is equivalent to a property stronger than the MIP. As corollaries, we recapture the result of Lau and characterize the w*-MIP in dual of RNP spaces.
Abstract. A subspace Y of a Banach space X is an almost constrained (AC) subspace if any family o... more Abstract. A subspace Y of a Banach space X is an almost constrained (AC) subspace if any family of closed balls centred at points of Y that intersects in X also intersects in Y. In this paper, we show that a subspace H of finite codimension in C(K), the space of continuous functions on a compact Hausdorff space K, is an AC-subspace if and only if H is the range of a norm one projection in C(K). We also give a simple proof that the implication “AC ⇒ 1-complemented ” holds for any subspace of c0(Γ) and c. 1.
Prasanta Chandra Mahalanobis—A Scientist and an Entrepreneur
The man in the making Prasanta Chandra Mahalanobis was one of the last generation of men and wome... more The man in the making Prasanta Chandra Mahalanobis was one of the last generation of men and women whose life and times were Iargely shaped by the traditions of the Bengal Renaissance .
Almost isometric ideals and non-separable Gurariy spaces
Journal of Mathematical Analysis and Applications
We work with real Banach spaces. We will denote by B(X), S(X) and B[x, r] respectively the closed... more We work with real Banach spaces. We will denote by B(X), S(X) and B[x, r] respectively the closed unit ball, the unit sphere and the closed ball of radius r > 0 around x ∈ X. We will identify any element x ∈ X with its canonical image in X∗∗. All subspaces we usually consider are norm closed. Definition 1.1. (a) We say A ⊆ B(X∗) is a norming set for X if ‖x‖ = sup{x∗(x) : x∗ ∈ A}, for all x ∈ X. A closed subspace F ⊆ X∗ is a norming subspace if B(F ) is a norming set for X. (b) A Banach space X is (i) nicely smooth if X∗ contains no proper norming subspace; (ii) has the Ball Generated Property (BGP) if every closed bounded convex set in X is ball-generated, i.e., intersection of finite union of balls; (iii) has Property (II) if every closed bounded convex set in X is the intersection of closed convex hulls of finite union of balls, or equivalently, w*-points of continuity (w*-PCs) of B(X∗) are norm dense in S(X∗) [5]; (iv) has the Mazur Intersection Property (MIP) (or, Property (...
Illinois Journal of Mathematics
We study the abstract geometric notion of unitaries in a Banach space characterized in terms of t... more We study the abstract geometric notion of unitaries in a Banach space characterized in terms of the equivalence of the norm determined by the state space.
Colloquium Mathematicum
Introduction. S. Mazur [10] was the first to consider the following smoothness property in normed... more Introduction. S. Mazur [10] was the first to consider the following smoothness property in normed linear spaces, called the Mazur Intersection Property (MIP), or, more briefly, the Property (I): Every closed bounded convex set is the intersection of closed balls containing it.
Taiwanese Journal of Mathematics
In this survey article, we explore various natural situations where the results about strict conv... more In this survey article, we explore various natural situations where the results about strict convexity of X ¤ like Vlasov's Theorem or Taylor-Foguel Theorem are actually seen to be locally consequences of properties of rotund points.
Journal of Convex Analysis, 2006
We show, among other results, that if the unit ball of the dual of a Banach space X is w * -seque... more We show, among other results, that if the unit ball of the dual of a Banach space X is w * -sequentially compact, the set of normattaining functionals contains a separable norm closed subspace M if and only if the dual M * of M is the canonical quotient of X. We provide examples of spaces which cannot be renormed in such a way that the set of norm-attaining functionals become a linear space.
Extracta Mathematicae, 2007
Let be a linear functional on a subspace Y of a real linear space X provided with a sublinear fun... more Let be a linear functional on a subspace Y of a real linear space X provided with a sublinear functional p with ≤ p on Y . If G is an abelian semigroup of linear transformations T : X → X such that T (Y ) ⊆ Y , p(T x) ≤ p(x) and (T y) = (y) for all T ∈ G, x ∈ X and y ∈ Y respectively, then a generalization of the classical Hahn-Banach theorem asserts that there exists an extension of , ≤ p on X and remains invariant under G. The present paper investigates various equivalent conditions for the uniqueness of such extensions and these are related to nested sequences of p-balls, a concept that has proved useful in recent years in dealing with such extensions. The results are illustrated by a variety of examples and applications.
Extreme contractions in ℒ(ℓ 2 p ,ℓ 2 q ) and the Mazur intersection property in ℓ 2 p ⊗ π ℓ 2 q
Rotund points, nested sequence of balls and smoothness in Banach spaces
K.\ S.\ Lau had shown that a reflexive Banach space has the Mazur Intersection Property (MIP) if ... more K.\ S.\ Lau had shown that a reflexive Banach space has the Mazur Intersection Property (MIP) if and only if every closed bounded convex set is the closed convex hull of its farthest points. In this work, we show that in general this latter property is equivalent to a property stronger than the MIP. As corollaries, we recapture the result
In this work, we obtain some necessary and some sufficient conditions for a space to be nicely sm... more In this work, we obtain some necessary and some sufficient conditions for a space to be nicely smooth, and show that they are equivalent for separable or Asplund spaces. We obtain a sufficient condition for the Ball Generated Property (BGP), and conclude that Property $(II)$ implies the BGP, which, in turn, implies the space is nicely smooth. We show that
In this paper, we obtain some sufficient conditions for an almost constrained subspace to be cons... more In this paper, we obtain some sufficient conditions for an almost constrained subspace to be constrained (in fact, by a unique norm 1 projection), which improves significantly upon all existing conditions of similar type with significantly simpler proofs.
Contemporary Mathematics, 2003
Veselý has studied Banach spaces that admit weighted Chebyshev centres for finite sets. Subsequen... more Veselý has studied Banach spaces that admit weighted Chebyshev centres for finite sets. Subsequently, Bandyopadhyay and Rao had shown, inter alia, that L 1-preduals have this property. In this work, we investigate why and to what extent are these results true and thereby explore when a more general family of sets admit weighted Chebyshev centres. We extend and improve upon some earlier results in this general setup and relate them with a modified notion of minimal points. Special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of L 1-preduals. We also consider some stability results.
Positivity, 2009
In this paper, we show that in many function spaces and function algebras, the unit ball of any M... more In this paper, we show that in many function spaces and function algebras, the unit ball of any M-ideal is densely remotal.
Exposed points and points of continuity in closed bounded convex sets
Lecture Notes in Mathematics, 1992
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Papers by Pradipta Bandyopadhyay