Some geometric properties of spaces of vector-valued integrable functions

TL;DR

This paper investigates the geometric structure of spaces of vector-valued integrable functions, focusing on smooth points and symmetry properties related to Birkhoff-James orthogonality in various L^p spaces.

Contribution

It characterizes smooth points and provides conditions for orthogonality symmetry in vector-valued L^p spaces, extending geometric understanding of these function spaces.

Findings

Characterization of smooth points in L^1(,X)

Necessary and sufficient conditions for orthogonality symmetry in L^p(,X)

Extension of geometric properties to spaces with general measures

Abstract

We identify the smooth points of $L^1(\mu,X)$, and provide some necessary and sufficient conditions for left and right symmetry of points with respect to Birkhoff-James orthogonality in $L^p(\mu,X), 1\leq p<\infty$, where $\mu$ is any complete positive measure and $X$ is a Banach space with some suitable properties.