Stability in vector valued \(\ell^ \infty\)-spaces. (English) Zbl 0785.46028

Graślewicz, Ryszard

Acta Univ. Carol., Math. Phys. 33, No. 2, 41-44 (1992).

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