Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Linear Hahn–Banach extension operators

Brailey Simsa1 and David Yosta2

a1 Department of Mathematics, Statistics and Computing Science University of New England Armidale N.S.W. 2351 Australia

a2 Department of Mathematics Institute of Advanced Studies Australian National University G.P.O. Box 4 A.C.T. 2601 Australia

Given any subspace N of a Banach space X, there is a subspace M containing N and of the same density character as N, for which there exists a linear Hahn–Banach extension operator from M* to X*. This result was first proved by Heinrich and Mankiewicz [4, Proposition 3.4] using some of the deeper results of Model Theory. More precisely, they used the Banach space version of the Löwenheim–Skolem theorem due to Stern [11], which in turn relies on the Löwenheim–Skolem and Keisler–Shelah theorems from Model Theory. Previously Lindenstrauss [7], using a finite dimensional lemma and a compactness argument, obtained a version of this for reflexive spaces. We shall show that the same finite dimensional lemma leads directly to the general result, without any appeal to Model Theory.

(Received May 05 1987)