Computability of the Hahn-Banach Theorem Revisited

TL;DR

This paper investigates the computational complexity of the Hahn-Banach theorem, showing it attains full complexity in specific Banach spaces and establishing equivalences with well-known principles in computability theory.

Contribution

It proves that the Hahn-Banach theorem for ll^1 reaches its full Weihrauch complexity and establishes equivalences with the intermediate value theorem and lesser limited principle of omniscience.

Findings

Hahn-Banach theorem for ll^1 has full Weihrauch complexity.

One-step Hahn-Banach for ll^1 is equivalent to the intermediate value theorem.

Hahn-Banach for ll^1 in two dimensions is equivalent to the lesser limited principle of omniscience.

Abstract

Computational properties of the Hahn-Banach theorem have been studied in computable, constructive and reverse mathematics and in all these approaches the theorem is equivalent to weak K\H{o}nig's lemma. Gherardi and Marcone proved that this is also true in the uniform sense of Weihrauch complexity. However, their result requires the underlying space to be variable. We prove that the Hahn-Banach theorem attains its full complexity already for the Banach space $\ell^1$. We also prove that the one-step Hahn-Banach theorem for this space is Weihrauch equivalent to the intermediate value theorem. This also yields a new and very simple proof of the reduction of the Hahn-Banach theorem to weak K\H{o}nig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for $\ell^1$ in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience.