Complemented subspaces of locally convex direct sums of Banach spaces

Abstract

We show that a complemented subspace of a locally convex direct sum of an uncountable collection of Banach spaces is a locally convex direct sum of complemented subspaces of countable subsums. As a corollary we prove that a complemented subspace of a locally convex direct sum of arbitrary collection of $\ell_{1}(\Gamma)$-spaces is isomorphic to a locally convex direct sum of $\ell_{1}(\Gamma)$-spaces.