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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
##### Keywords:
convex subset; midpoint map; stability of the unit balls
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