John Y. Barry
Feb 1, 1954
Journal name not available for this finding
E. R. Lorch has shown [3, p. 223 ] that a uniformly bounded, naturally ordered sequence of projections in a reflexive Banach space converges to its supremum in the strong topology of operators. In this note it is shown that a uniformly bounded, naturally ordered set of projections in a Banach space X having a weak x-cluster point for each x CX converges to its supremum in the strong topology of operators. The method of proof is substantially different from that of Lorch. A bounded operator E, acting in a Banach space X, is a projection if E2 =E. Letting I denote the identity projection, E(X) and (I-E) (X) are strongly closed manifolds. The projections in X have a natural order: E?