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On Hahn-Banach smoothness and related properties in Banach spaces

Soumitra Daptari

Abstract

In this paper, we study several variants of Hahn-Banach smoothness, viz., property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed that $L_1(μ)$ has property-$(wU)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ but it does not have property-$(U)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ for a non-atomic measure $μ$. We derive a sufficient condition when property-$(wU)$ is equivalent to property-$(U)$ of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of $c_0$, $\ell_p$ ($1\leq p<\infty$) having these properties are characterized.

On Hahn-Banach smoothness and related properties in Banach spaces

Abstract

In this paper, we study several variants of Hahn-Banach smoothness, viz., property-//, where property- and property- are stronger notions and property- is a weaker notion of Hahn-Banach smoothness. We characterize property- and property-. It is observed that has property- in but it does not have property- in for a non-atomic measure . We derive a sufficient condition when property- is equivalent to property- of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of , () having these properties are characterized.
Paper Structure (10 sections, 33 theorems, 1 equation)

This paper contains 10 sections, 33 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. A short background and motivation of this work
  4. Property-$(U)$ and its alterations
  5. On property-$(wU)$
  6. On property-$(HB)$
  7. Separable determination of these properties.
  8. On Hahn-Banach smoothness and related properties in sequence spaces
  9. In finite-dimensional subspaces
  10. In finite co-dimensional subspaces

Key Result

Theorem 2.1

Let $Y$ be a subspace of $X$. Then, the following are equivalent:

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 2.3
  • Theorem 2.4
  • proof
  • ...and 47 more