Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 547–564 | Cite as

Borel complexity and computability of the Hahn–Banach Theorem

Article

Abstract

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is \({\bf {\Sigma^{0}_{2}}}\) -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions.

Keywords

Computable analysis Functional analysis Effective descriptive set theory \({\bf \Pi^{0}_{1}}\) -sets 

Mathematics Subject Classification (2000)

03F60 03E15 46S30 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Laboratory of Foundational Aspects of Computer Science, Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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