Relative bases in Banach spaces

We give, in this work, a new basis definition for Banach spaces and investigate some structural properties of certain vector-valued function spaces by using it. By novelty of the new definition, we prove that l(infinity) has a basis in this sense, and so we deduce as a result that it has approximation property. In fact, we obtain a more general result that the linear subspace P (B, X) of l(infinity) (B, X) of all those functions with a precompact range has an XSchauder basis. Hence P (A, X) has approximation property if and only if the Banach space X has. Note that P (B, X) = l(infinity) (B, X) for some finite-dimensional X. Further, we give a representation theorem to operators on certain vector-valued function spaces. (C) 2009 Elsevier Ltd. All rights reserved.