Uniqueness of Hahn–Banach extensions and some of its variants

Daptari, Soumitra and Paul, Tanmoy (2022) Uniqueness of Hahn–Banach extensions and some of its variants. Advances in Operator Theory, 7 (3). ISSN 2662-2009

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Abstract

In this paper, we analyze the various strengthening and weakening of the uniqueness of the Hahn–Banach extension. In addition, we consider the case in which Y is an ideal of X. In this context, we study the property (U)/(SU)/(HB) and property (wU)/(k-U) for a subspace Y of a Banach space X. We obtain various new characterizations of these properties. We study different kinds of stabilities resulting from these properties in the tensor product spaces, spaces of Bochner integrable functions, and the higher duals of Banach spaces. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is established that a hyperplane in c has property (HB) if and only if it is an M-summand. It is observed that a finite-dimensional subspace Y has property (k-U) in c, in addition to that if Y is an ideal, then Y∗ is a k-strictly convex subspace of ℓ1 for some natural k. © 2022, Tusi Mathematical Research Group (TMRG).

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