On the Strength of Weak Compactness

Article type: Research Article

Authors: Kreuzer, Alexander P.^;

Affiliations: LIP (UMR 5668), ENS Lyon, 46 Allée d'Italie, 69364 Lyon Cedex 07, France. alexander.kreuzer@ens-lyon.fr

Note: [] The author is supported by the German Science Foundation (DFG Project KO 1737/5-1). I am grateful to Ulrich Kohlenbach for useful discussions and suggestions for improving the presentation of the material in this article. I would like to thank the anonymous referees for corrections and suggestions that have helped to improve the revised version of this article.

Abstract: We study the logical and computational strength of weak compactness in the separable Hilbert space ℓ2. Let weak-BW be the statement the every bounded sequence in ℓ2 has a weak cluster point. It is known that weak-BW is equivalent to ACA0 over RCA0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstraß principle BW. We show that weak-BW is instance-wise equivalent to $\Pi^0_2$-CA. This means that for each $\Pi^0_2$ sentence A(n) there is a sequence $(x_i)_{i\in{\mathbb{N}}})$ in ℓ2, such that one can define the comprehension function for A(n) recursively in a cluster point of (xi)i. As a consequence we obtain that the degrees d≥T 0″ are exactly the degrees that contain a weak cluster point of any computable, bounded sequence in ℓ2. Since a cluster point of any sequence in the unit interval [0,1] can be computed in a degree low over 0′ (see [10]), this also shows that instances of weak-BW are strictly stronger than instances of BW. We also comment on the strength of weak-BW in the context of abstract Hilbert spaces in the sense of Kohlenbach and show that his construction of a solution for the functional interpretation of weak compactness is optimal, cf. [7]. 2010 Mathematics Subject Classification. Primary 03F60; Secondary 03D80, 03B30.

Keywords: Bolzano-Weierstraß principle, weak sequential compactness, Turing degree, abstract Hilbert space

DOI: 10.3233/COM-12010

Journal: Computability, vol. 1, no. 2, pp. 171-179, 2012