Separable Banach space theory
needs strong set existence axioms

A. James Humphreys
Stephen G. Simpson

simpson@math.psu.edu
Pennsylvania State University
May 17, 1995

This research was partially supported by NSF grant DMS-9303478. We would also like to thank our colleague Robert E. Huff for showing us his unpublished notes on the Krein-Smulian theorem, and the referee for helpful comments which improved the exposition of this paper.

Subject classification: Primary 03F35; Secondary 46B10, 46B45

Keywords: Reverse mathematics, separable Banach space theory, weak-*topology, closure ordinals, Krein-Smulian theorem.

Published in Transactions of the American Mathematical Society, 348, 1996, pp. 4231-4255.

Abstract:

We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, $\Pi^1_1$ comprehension, is needed to prove such basic facts as the existence of the weak-* closure of any norm-closed subspace of $\ell_1=c_0^*$. This is in contrast to earlier work [5,7,6,23,22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for $\Pi^0_2$ sentences. En route to our main results, we prove the Krein-Smulian theorem in $\mathsf{ACA}_0$, and we give a new, elementary proof of a result of McGehee on weak-* sequential closure ordinals.