Completeness of Sums of Subspace of Bounded Functions and Applications

TL;DR

This paper provides a new proof for characterizing the closeness of the range of linear operators and sums of subspaces in Banach spaces, with applications to spaces of bounded functions and almost periodic functions.

Contribution

It introduces a novel proof technique for subspace closeness and establishes sufficient conditions for sums of subspaces in bounded function spaces, with practical applications.

Findings

New proof of subspace closeness characterization

Sufficient conditions for sum closeness in bounded function spaces

Applications to pseudo almost periodic and automorphic spaces

Abstract

We give a new proof of a characterization of the closeness of the range of a continuous linear operator and of the closeness of the sum of two closed vector subspaces of a Banach space. Then we state sufficient conditions for the closeness of the sum of two closed subspaces of the Banach space of bounded functions and apply this result on various pseudo almost periodic spaces and pseudo almost automorphic spaces.