On the theorem of Helley concerning finite-dimensional subspaces of a dual space

James Eugene Shirey

Vol. 86 (1980), No. 2, 571–577

DOI: 10.2140/pjm.1980.86.571

Abstract

Let C and S denote Banach spaces for which C ⊂ S^∗. Then (S,C) is said to have property [P] if for any s^∗∗∈ S^∗∗ there is an s ∈ S such that s^∗∗(c) = c(s) for every c ∈ C. There is then a fixed M > 0 so that if 𝜖 > 0, s can always be chosen so that ∥s∥≦ M∥s^∗∗∥ + 𝜖. If M = 1, then (S,C) is a 1-Helley pair.

A classical theorem of E. Helley states that (S,C) is a 1-Helley pair whenever C is finite dimensional. It is shown that such is the case whenever C is reflexive. As a partial converse, if (S,C) has property [P] and if C is weak* closed, then C is reflexive.

It is also shown that if X and Y are closed subspaces of a Banach space, and if X + Y is closed, then there is M > 0 so that for each z ∈ X + Y , there are x ∈ X and y ∈ Y for which z = x + y and ∥x∥≦ M∥z∥.

Mathematical Subject Classification 2000

Primary: 46B10

Secondary: 52A07, 52A35

Milestones

Received: 12 December 1977

Revised: 10 January 1979

Published: 1 February 1980