needs strong set existence axioms

**A. James Humphreys
Stephen G. Simpson**

`t20@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 We would also like to thank our colleague Robert E. Huff for showing us his unpublished notes on the Krein-Smulian theorem, and the referee for helpful comments which improved the exposition of this paper.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.

We investigate the strength of set existence axioms needed for
separable Banach space theory. We show that a very strong axiom,
comprehension, is needed to prove such basic facts as the
existence of the weak-* closure of any norm-closed subspace of
.
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
sentences. En route to our main
results, we prove the Krein-Smulian theorem in
,
and we give a
new, elementary proof of a result of McGehee on weak-* sequential
closure ordinals.

- Introduction
- Banach space preliminaries
- Trees and subspaces of
- The weak-* topology in subsystems of
*Z*_{2} - Proof of the main result
- Bibliography
- About this document ...