Completeness of Sums of Subspace of Bounded Functions and Applications

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.