Vol. 33, No. 2, 1970

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Sections and subsets of simplexes

Aldo Joram Lazar

Vol. 33 (1970), No. 2, 337–344

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

S = {(an) :  an = 1,an ≧ 0,n = 1,2,⋅⋅⋅}.

If X is a Banach space and S X is a compact simplex, then for each 𝜖 > 0 there is an operator T : X X with finite dimensional range such that T(x) x< 𝜖 for all x S. Every infinite dimensional Banach space X contains a compact set K for which there is no bounded simplex S X with K S.

Mathematical Subject Classification
Primary: 46.01
Received: 2 July 1969
Revised: 10 November 1969
Published: 1 May 1970
Aldo Joram Lazar