Notre Dame Journal of Formal Logic

How Incomputable Is the Separable Hahn-Banach Theorem?

Guido Gherardi and Alberto Marcone


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 König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete.

Article information

Notre Dame J. Formal Logic, Volume 50, Number 4 (2009), 393 - 425 .

First available in Project Euclid: 11 February 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F60
Secondary: 03B30 46A22 46S30

computable analysis reverse mathematics weak Konig's lemma Hahn-Banach extension theorem multivalued functions


Gherardi , Guido; Marcone , Alberto. How Incomputable Is the Separable Hahn-Banach Theorem?. Notre Dame J. Formal Logic 50 (2009), no. 4, 393 -- 425. doi:10.1215/00294527-2009-018.

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