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arXiv · 2202.11373

Hilbert points in Hilbert space-valued $L^p$ spaces

Ole Fredrik Brevig·Sigrid Grepstad
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Abstract

Let $H$ be a Hilbert space and $(\Omega,\mathcal{F},\mu)$ a probability space. A Hilbert point in $L^p(\Omega; H)$ is a nontrivial function $\varphi$ such that $\|\varphi\|_p \leq \|\varphi+f\|_p$ whenever $\langle f, \varphi \rangle = 0$. We demonstrate that $\varphi$ is a Hilbert point in $L^p(\Omega; H)$ for some $p\neq2$ if and only if $\|\varphi(\omega)\|_H$ assumes only the two values $0$ and $C>0$. We also obtain a geometric description of when a sum of independent Rademacher variables is a Hilbert point.

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BibTeXRIS

Ole Fredrik Brevig, Sigrid Grepstad. 2022-02-23. Hilbert points in Hilbert space-valued $L^p$ spaces. https://arxiv.org/abs/2202.11373

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