Remarks on quasi-Banach spaces over valued fields
Let \(K\) be a non-Archimedean valued field that is complete with respect to the nontrivial valuation \(|\cdot|\). Like in functional analysis over \(\mathbb{R}\) or \(\mathbb{C}\) one introduces the notion of a quasinorm on a \(K\)-vector space \(E\) by relaxing the triangle inequality to \(\| x+y\|\leq k(\| x\|+\| y\|)\) for some constant \(k\geq 1\). The Mackey envelope \(\mu\) of a quasinormed space \((E,\tau)\) is by definition the finest locally convex topology \(\leq\tau\). It is shown that, if \((E,\tau)\) is dual-separating and of countable type, then \((E,\mu)\) is a normed space whose completion is isomorphic to \(c_0\). Several conditions on a quasi-Banach space \((E,\tau)\) in order that \(\tau= \mu\) are presented (such as: every null sequence is summable; every weak Schauder base is a basic sequence and the adjoint \(E'\to D'\) is surjective for each subspace \(D\hookrightarrow E\)). As examples the spaces \(\ell^p\) \((0<p<1)\) are being discussed.
scientific article; zbMATH DE number 1123836