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∞ℓ∞ 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)P(B,X) of ℓ∞(B,X)ℓ∞(B,X) of all those functions with a precompact range has an XX-Schauder basis. Hence P(A,X)P(A,X) has approximation property if and only if the Banach space XX has. Note that P(B,X)=ℓ∞(B,X)P(B,X)=ℓ∞(B,X) for some finite-dimensional XX. Further, we give a representation theorem to operators on certain vector-valued function spaces.