zbMATH — the first resource for mathematics

On a class of free locally convex spaces. (English. Russian original) Zbl 1079.54014
Sb. Math. 194, No. 3, 333-360 (2003); translation from Mat. Sb. 194, No. 3, 25-52 (2003).
In 1951, Dugundji proved a celebrated extension theorem. In 1966, Borges extended Dugundji’s theorem from metrizable spaces to a larger class, the stratifiable spaces. A space \(X\) is said to be stratifiable if each closed subset \(F\) of it can be represented in the following form: \[ F=\bigcap^\infty_{n=1}U_n(F)=\bigcap^\infty_{n=1}\overline{U_n(F)} \] with all the \(U_n(F)\) open, so \(U_n(F')\subset U_n(F)\) for all \(n\) whenever \(F'\) is a closed subset of \(F\). Let \(X\) be a topological space. Its free locally convex space \(L(X)\) is the free real vector space generated by \(X\), that is, the set of formal finite linear combinations of elements of \(X\) with real coefficients, endowed with the finest locally convex topology inducing the original topology on \(X\). The free locally convex space of a stratifiable \(T_1\)-space is stratifiable. The proof of this theorem is very laborious. This result and the Dugundji-Borges theorem yield the following two results: (a) The space of finitely supported probability measures on a stratifiable space is a retract of a locally convex space. (b) Every stratifiable convex subset of a locally convex space is a retract of a locally convex space.

54E20 Stratifiable spaces, cosmic spaces, etc.
46A03 General theory of locally convex spaces
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
54C15 Retraction
Full Text: DOI