Characterizing Hilbert spaces using Fourier transform over the field of p-adic numbers

TL;DR

This paper characterizes Hilbert spaces among Banach spaces by analyzing the boundedness of the Fourier transform over the p-adic number field, providing a new perspective on their structure.

Contribution

It introduces a novel characterization of Hilbert spaces via Fourier transform boundedness over the p-adic field, linking harmonic analysis with Banach space theory.

Findings

Banach space is Hilbert if Fourier transform on p-adic functions is bounded

Provides a new criterion for identifying Hilbert spaces

Connects harmonic analysis with Banach space geometry

Abstract

We characterize Hilbert spaces in the class of all Banach spaces using Fourier transform of vector-valued functions over the field $Q_p$ of $p$-adic numbers. Precisely, Banach space $X$ is isomorphic to a Hilbert one if and only if Fourier transform $F: L_2(Q_p,X)\to L_2(Q_p,X)$ in space of functions, which are square-integrable in Bochner sense and take value in $X$, is a bounded operator.