Some geometric properties of spaces of vector-valued integrable functions

TL;DR

This work analyzes the geometric structure of vector-valued Lebesgue-Bochner spaces under Birkhoff-James orthogonality. By assuming a Fréchet-differentiable norm on the underlying Banach space and examining the measure-theoretic atom structure of the zero-set complement $Z(f)^c$, it precisely characterizes smooth points in $L^{1}(\mu,X)$ as exactly the functions with $f\neq0$ a.e., and establishes necessary and sufficient conditions for left and right symmetry in $L^{1}(\mu,X)$ and in $L^{p}(\mu,X)$ for $1<p<\infty$, $p\neq2$. The results reveal that left symmetry forces $Z(f)^c$ to be an atom or a two-atom union (with partial converses and sufficiency results), while non-atomic measures eliminate nonzero symmetric points in these spaces. Collectively, the findings extend the understanding of local geometric properties from scalar to vector-valued function spaces and illuminate how measure-atomic structure governs symmetry phenomena in $L^{p}(\mu,X)$.

Abstract

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

Theorems & Definitions (33)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof
• Theorem 3.4
• proof
• Corollary 3.5