Abstract
LetK be a locally compact non-archimedean non-trivially valued field. It is proved the theorem: For a Banach space overK containing a dense subspace with the Hahn-Banach extension property one of the following two mutually exclusive conditions holds:E is a non-archimedean Banach space or the space {x∈E:f(x)=0 for allf∈E*} has no non-trivial continuous linear functionals. Two corollaries are also obtained.
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