K-Smooth Points in Some Banach Spaces

A Saleh Hamarsheh

doi:10.1155/2015/394282

Abstract

We characterize thek-smooth points in some Banach spaces. We will deal with injective tensor product, the Bochner spaceL∞(μ,X)of (equivalence classes of)μ-essentially bounded measurableX-valued functions, and direct sums of Banach spaces.

Highlights

For a unit vector x in a Banach space X, consider the state space Sx = {x∗ ∈ X∗ : ‖x∗‖ = 1 = x∗(x)}
The set of all smooth points is denoted by smooth B(X)
Multismoothness in Banach spaces was extensively studied by Lin and Rao in [9]

Summary

Introduction

For two Banach spaces X, Y Heinrich, [1], gave a description of smooth points of the unit ball in the space K(X, Y) of compact operators from X into Y. In [8] the authors generalize the notion of smoothness by calling a unit vector x in a Banach space X a k-smooth point, or a multismooth point of order k if Sx has exactly k linearly independent vectors, equivalently, if dim(sp Sx) = k. Multismoothness in Banach spaces was extensively studied by Lin and Rao in [9] In paricular, they showed that, in a Banach space of finite dimension k, any k-smooth point is unitary and a strongly extreme point. The set of all extreme points of the unit ball of a Banach space X is denoted by ext B(X)

Multismoothness in Injective Tensor Products

Multismoothness in Bochner Spaces

Multismoothness in Direct Sums of Banach Spaces