6533b86cfe1ef96bd12c89d4

RESEARCH PRODUCT

Topological direct sum decompositions of banach spaces

M. Valdivia

subject

Discrete mathematicsDense setDirect sumGeneral MathematicsExistential quantificationBanach spaceBanach manifoldAlgebra over a fieldTopologyLinear subspaceMathematics

description

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T −1(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX 1 andX 2 such thatX 1 is reflexive and densX 2**=densX**/X.

https://doi.org/10.1007/bf02773747