How incomputable is the separable Hahn-Banach theorem?

Details How incomputable is the separable Hahn-Banach theorem? How incomputable is the separable Hahn-Banach theorem?

2008-08-12

arxiv.org/abs/0808.1663v3

Notre Dame Journal of Formal Logic 50 (2009) 393-425

Mathematics

Logic

final version to appear in Notre Dame Journal of Formal Logic

Scientific paper

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak Konig's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL_0. We study analogies and differences between WKL_0 and the class of Sep-computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

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