K\"othe-Bochner spaces and some geometric properties related to rotundity and smoothness

TL;DR

This paper explores how geometric properties related to rotundity and smoothness in Banach spaces behave when forming K"othe-Bochner spaces, extending previous results on $L^p$-Bochner spaces.

Contribution

It generalizes the understanding of rotundity and smoothness properties in K"othe-Bochner spaces, building on prior work on absolute sums and $L^p$-spaces.

Findings

Properties are preserved under K"othe-Bochner space formation

Generalizes results from $L^p$-Bochner spaces to broader K"othe-Bochner spaces

Provides new insights into geometric properties of Banach spaces

Abstract

In 2000 Kadets et al. introduced the notions of acs, luacs and uacs spaces, which form common generalisations of well-known rotundity and smoothness properties of Banach spaces. In a recent preprint the author introduced some further related notions and investigated the behaviour of these geometric properties under the formation of absolute sums. This paper is in a sense a continuation of the previous work. Here we will study the behaviour of said properties under the formation of K\"othe-Bochner spaces, thereby generalising some results of Sirotkin on the acs, luacs and uacs properties of $L^p$-Bochner spaces.