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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.

##### MSC:
 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
##### Keywords:
stratifiable space; probability measures; retract
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