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Weak and strong \(c'\)-compactness in non-Archimedean Banach spaces. (English) Zbl 0812.46077

Summary: Throughout \({\mathbf K}\) is a non-archimedean complete valued field with dense valuation \(|\cdot|\). An absolutely convex set \(A\) of a \({\mathbf K}\)-Banach space \(E\) is called (weakly) \(c'\)-compact if \(\max_{x\in A} p(x)\) exists for each (weakly) continuous seminorm \(p\) on \(E\). Assuming the continuum hypothesis, we prove that, if \({\mathbf K}\) has the cardinality of the continuum, in a strongly polar \({\mathbf K}\)- Banach space, each weakly \(c'\)-compact set is \(c'\)-compact.

MSC:

46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
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