Correction of a generalization of a theorem of Beurling and Livingston

Abstract:

A generalization of a Riesz-Fischer theorem proved by Beurling and Livingston for smooth uniformly convex Banach spaces also holds for smooth, strictly convex, reflexive Banach spaces. Theorem. Let B be a smooth, strictly convex, reflexive Banach space. Let $L:B \to {B^\ast }$ be a duality map, C a closed subspace of $B,h \in B,k \in {B^\ast }$. Then $T(C + h) \cap ({C^ \bot } + k)$ is a single point. A two-dimensional counterexample shows that $T(C + h) \cap ({C^ \bot } + k) = \emptyset$ is possible if B is not smooth, contrary to the claim of Theorem 4 of Browder, On a theorem of Beurling and Livingston, Canad. J. Math. 17 (1965), 367-372.