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

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.

DOI: 10.3336/gm.49.2.13

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