Estimates for projections in Banach spaces and existence of direct complements

Abstract

Let $W$ and $L$ be complementary subspaces of a Banach space $X$ and let $P(W,L)$ denote the projection on $W$ along $L$. We obtain a sufficient condition for a subspace $M$ of $X$ to be complementary to $W$ and we derive estimates for the norm of $P(W,L)