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Computability of the Hahn-Banach Theorem Revisited

Vasco Brattka, Christopher Sorg

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 Kőnig'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 Kőnig'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.

Computability of the Hahn-Banach Theorem Revisited

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 Kőnig'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 . 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 Kőnig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience.
Paper Structure (7 sections, 26 theorems, 34 equations, 2 figures)

This paper contains 7 sections, 26 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Weihrauch Complexity and the Hahn-Banach Theorem
  3. The One-Step Hahn-Banach Theorem
  4. The Hahn-Banach Theorem for $\ell^1_2$
  5. The Hahn-Banach Theorem for $\ell^1$
  6. The Hahn-Banach Theorem for Located Subspaces

Key Result

theorem 1

Let $X$ be a normed space with a linear subspace $A\subseteq X$. Then every linear bounded functional $f:A\to{\mathbb{R}}$ has a linear extension $g:X\to{\mathbb{R}}$ with $\|g\|=\|f\|$.

Figures (2)

  • Figure 1: The Hahn-Banach theorem in the Weihrauch lattice.
  • Figure 2: Unit balls in ${\mathbb{R}}^2$ with respect to $\ell^1$ and $\ell^\infty$.

Theorems & Definitions (39)

  • theorem 1: Hahn-Banach
  • theorem 2: Gherardi and Marcone 2009
  • corollary 1: Metakides, Nerode and Shore 1985
  • corollary 2
  • theorem 3
  • theorem 4: One-step Hahn-Banach theorem
  • proposition 1
  • theorem 5
  • proposition 2
  • proposition 3
  • ...and 29 more