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

On completeness of the space of weighted pseudo almost automorphic functions