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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Separable Banach space theory
needs strong set existence axioms


Authors: A. James Humphreys and Stephen G. Simpson
Journal: Trans. Amer. Math. Soc. 348 (1996), 4231-4255
MSC (1991): Primary 03F35; Secondary 46B10, 46B45
MathSciNet review: 1373639
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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 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-\v{S}mulian theorem in $\mathsf {ACA}_0$, and we give a new, elementary proof of a result of McGehee on weak-$*$ sequential closure ordinals.


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Additional Information

A. James Humphreys
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvannia 16802
Email: jimbo@math.psu.edu

Stephen G. Simpson
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvannia 16802
Email: simpson@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01725-4
PII: S 0002-9947(96)01725-4
Keywords: Reverse mathematics, separable Banach space theory, weak-$*$ topology, closure ordinals, Krein-\v Smulian theorem
Received by editor(s): July 10, 1995
Additional Notes: 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 We would also like to thank our colleague Robert E. Huff for showing us his unpublished notes on the Krein-Šmulian theorem, and the referee for helpful comments which improved the exposition of this paper.
Article copyright: © Copyright 1996 American Mathematical Society