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Stability in vector valued \(\ell^ \infty\)-spaces. (English) Zbl 0785.46028
Summary: A convex subset \(Q\) of topological vector space is called stable if the midpoint map \(Q\times Q\ni (x,y)\to {1\over 2}(x+y)\in Q\) is open with respect to the inherited topology in \(Q\). The purpose of the paper is to discuss stability of the unit balls of spaces \(\ell^ \infty(E)\) and \({\mathcal L}(\ell^ 1,E)\) (\(E\) is a Banach space).
MSC:
46B45 Banach sequence spaces
46A45 Sequence spaces (including Köthe sequence spaces)
46E40 Spaces of vector- and operator-valued functions
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