Glasnik Matematicki, Vol. 49, No. 2 (2014), 407-419.

THE METRIC APPROXIMATION PROPERTY IN NON-ARCHIMEDEAN NORMED SPACES

Cristina Perez-Garcia and Wilhelmus H. Schikhof

Department of Mathematics, Facultad de Ciencias, Universidad de Cantabria , Avda. de los Castros s/n, 39071, Santander , Spain
e-mail: perezmc@unican.es

Weezenhof 3607, 6536 HC Nijmegen, The Netherlands
e-mail: schikhof@upcmail.nl


Abstract.   A normed space E over a rank 1 non-archimedean valued field K has the metric approximation property (MAP) if the identity on E can be approximated pointwise by finite rank operators of norm 1. Characterizations and hereditary properties of the MAP are obtained. For Banach spaces E of countable type the following main result is derived: E has the MAP if and only if E is the orthogonal direct sum of finite-dimensional spaces (Theorem 4.9). Examples of the MAP are also given. Among them, Example 3.3 provides a solution to the following problem, posed by the first author in [8, 4.5]. Does every Banach space of countable type over K have the MAP?

2010 Mathematics Subject Classification.   46S10, 46B28.

Key words and phrases.   Non-archimedean normed spaces, pseudoreflexivity, metric approximation property, finite-dimensional decomposition.


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DOI: 10.3336/gm.49.2.13


References:

  1. G. Bachman, Introduction to p-Adic Numbers and Valuation Theory, Academic Press, New York, 1964.
    MathSciNet    

  2. P. G. Casazza, Approximation properties, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam (2001), 271-316.
    MathSciNet    

  3. P. Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), 309-317.
    MathSciNet     CrossRef

  4. G. Godefroy, The Banach space c0, Extracta Math. 16 (2001), 1-25.
    MathSciNet    

  5. A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), 1-140.
    MathSciNet    

  6. A. Kubzdela, On finite-dimensional normed spaces over  p, Contemp. Math. 384 (2005), 169-185.
    MathSciNet     CrossRef

  7. A. Kubzdela, On non-Archimedean Hilbertian spaces, Indag. Math. (N.S.) 19 (2008), 601-610.
    MathSciNet     CrossRef

  8. C. Perez-Garcia, Bounded approximation properties in non-archimedean Banach spaces, Math. Nach. 285 (2012), 1255-1263.
    MathSciNet     CrossRef

  9. C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over Non-Archimedean Valued Fields, Cambridge University Press, Cambridge, 2010.
    MathSciNet    

  10. A. M. Robert, A Course in p-Adic Analysis, Springer, Berlin, 2000.
    MathSciNet    

  11. W. H. Schikhof, Ultrametric Calculus. An Introduction to p-Adic Analysis, Cambridge University Press, Cambridge, 1984.
    MathSciNet    

  12. W. H. Schikhof, Banach spaces over nonarchimedean valued fields, Topology Proc. 24 (1999), 547-581.
    MathSciNet    

  13. A. C. M. van Rooij, Non-Archimedean Functional Analysis, Dekker, New York, 1978.
    MathSciNet    

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