Sections and subsets of simplexes

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There is a locally convex space E and a compact simplex S ⊂ E with the following property: for any metrizable compact convex subset K of a locally convex space there is a subspace M ⊆ E such that K is affinely homeomorphic to M ∩ S. One possible choice is E = l1 with the w topology induced by c and (Formula Presented) If X is a Banach space and S ⊂ X is a compact simplex, then for each s Ì 0 there is an operator T: X — X with finite dimensional range such that (Formula Presented) e for all x e S. Every infinite dimensional Banach space X contains a compact set K for which there is no bounded simplex S ⊂ X with K ⊂ S.

Original languageEnglish
Pages (from-to)337-344
Number of pages8
JournalPacific Journal of Mathematics
Issue number2
StatePublished - May 1970
Externally publishedYes


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