Computable Banach Spaces via Domain Theory

This paper extends the order-theoretic approach to computable analysis via continuous domains to complete metric spaces and Banach spaces. We employ the domain of formal balls to define a computability theory for complete metric spaces. For Banach spaces, the domain specialises to the domain of closed balls, ordered by reversed inclusion. We characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded. We show that the domain-theoretic computability theory is equivalent to the well-established approach by Pour-El and Richards.

[1]  Jürgen Hauck,et al.  Berechenbare Reelle Funktionen , 1973 .

[2]  Yu. L. Ershov.  Computable functionals of finite types , 1972 .

[3]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[4]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[5]  Y. Moschovakis.  Recursive metric spaces , 1964 .

[6]  Abbas Edalat,et al.  Domains for Computation in Mathematics, Physics and Exact Real Arithmetic , 1997, Bulletin of Symbolic Logic.

[7]  Abbas Edalat,et al.  A Domain-Theoretic Approach to Computability on the Real Line , 1999, Theor. Comput. Sci..

[8]  Christoph Kreitz,et al.  Theory of Representations , 1985, Theor. Comput. Sci..

[9]  Sören Stenlund.  Computable Functionals of Finite Type , 1972 .

[10]  Klaus Weihrauch.  A Foundation for Computable Analysis , 1997, SOFSEM.

[11]  Abbas Edalat,et al.  A Computational Model for Metric Spaces , 1998, Theor. Comput. Sci..

[12]  J. V. Tucker,et al.  Complete local rings as domains , 1988, Journal of Symbolic Logic (JSL).

[13]  Jens Blanck,et al.  Domain Representability of Metric Spaces , 1997, Ann. Pure Appl. Log..

[14]  Marian Boykan Pour-El,et al.  Computability in analysis and physics , 1989, Perspectives in Mathematical Logic.

[15]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[16]  Klaus Weihrauch,et al.  Computability on Computable Metric Spaces , 1993, Theor. Comput. Sci..

[17]  Abbas Edalat,et al.  A new representation for exact real numbers , 1997, MFPS.

[18]  Viggo Stoltenberg-Hansen,et al.  Concrete Models of Computation for Topological Algebras , 1999, Theor. Comput. Sci..

[19]  Jürgen Hauck.  Berechenbare Reelle Funktionenfolgen , 1976, Math. Log. Q..

[20]  J. V. Tucker,et al.  Effective algebras , 1995, Logic in Computer Science.