Characterizations of Schauder decompositions in Banach spaces

Abstract

The concept of a decomposition is a natural generalization of basis which was initiated in [5] and further studied in [3, 4, 8, 10, 11, 12]. Later on, in view of Enflo's example [2] which exhibits that every separable Banach space does not have an approximation property and hence no basis, the study of decompositions became more interesting and worth studying. It is worth noting that every separable Banach space do have a decomposition, but there are (non-separable) Banach spaces which do not have a Schauder decomposition; consider for instance the Banach space l=. Consequently, an attempt was made to obtain a criterion for the existence of Schauder decomposition of a Banach space and in this direction a theorem has been verified which corresponds to the theorem of Nikol'skii for the existence of bases [7]. Motivated by this work very recently Jain and Ahmad [6] obtained certain characterizations of Shcauder decompositions in terms of best approximations in Banach spaces. The purpose of this paper is to obtain some characterizations of Schauder decompositions in Banach spaces, which establish a new proof of the theorem of existence of Schauder decompositions of a Banach space (see [7], p. 93).