On completeness of the space of weighted pseudo almost automorphic functions

TL;DR

This paper proves the completeness of the space of weighted pseudo almost automorphic functions under the supremum norm, resolving a fundamental open problem and filling a gap in previous literature.

Contribution

It establishes the completeness of weighted pseudo almost automorphic functions for all weights in , addressing a key theoretical gap.

Findings

Confirmed the space's completeness under the supremum norm.

Provided an affirmative answer to a longstanding open problem.

Filled a gap in the proof from prior research.

Abstract

In this paper, we prove that for every $\rho\in \mathbb{U}_{\infty}$, the space of weighted pseudo almost automorphic functions is complete under the supremum norm. This gives an affirmative answer to a key and fundamental problem for weighted pseudo almost automorphic functions, and fills a gap in the proof of [J. Funct. Anal. 258, No. 1, 196-207 (2010)].