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

Completeness of Sums of Subspace of Bounded Functions and Applications