Date: May 17, 1995
Title: Separable Banach space theory needs strong set existence axioms
Authors: A. James Humphreys, Stephen G. Simpson
Author e-mail: simpson@math.psu.edu
Available: http://www.math.psu.edu/simpson/papers/convex.ps
Format: PostScript file (AMS-LaTeX source is also available)
Publication: Transactions of the AMS, 348, 1996, pp. 4231-4255.

Abstract: This paper is a contribution to the program of Reverse
Mathematics.  We investigate the role of strong set existence axioms
in separable Banach space theory.  Earlier work of Brown, Simpson,
Shioji and Tanaka has shown that many basic results of separable
Banach space theory are provable in weak systems such as WKL_0 and
RCA_0^+.  We now show that certain other basic results require a much
stronger system, Pi^1_1 comprehension.  Specifically, we show that
Pi^1_1 comprehension is necessary and sufficient to prove the
existence of the weak-* closure of a (norm-closed) subspace of the
dual of a separable Banach space.