Complexity theory of real functions
Complexity theory of real functions
Computable analysis: an introduction
Computable analysis: an introduction
The Inversion Problem for Computable Linear Operators
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
On Computable Metric Spaces Tietze-Urysohn Extension Is Computable
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
Computability on subsets of metric spaces
Theoretical Computer Science - Topology in computer science
On Computable Compact Operators on Banach Spaces
Electronic Notes in Theoretical Computer Science (ENTCS)
An analysis of the lemmas of urysohn and urysohn-tietze according to effective borel measurability
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
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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 on a compact metric space. This allows us to conclude that the Hahn-Banach extension operator is @?"2^0-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.