By J. L. Dupont, I. H. Madsen
Collage of Aarhus, 50. Anniversary, eleven September 1978
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Additional resources for Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978
5 of I, p. 8, to the case where the spaces Ei ( ~ i ~ n) and F are topological vector spaces over an arbitrary non-discrete topological field. 10) Let E be a complete Hausdorff topological vector space over a non-discrete valued division ring K. Denote by F a vector subspace of E, and by 'Y the topology on F induced by the topology 'Y' of E; let mbe a fundamental system of closed, balanced neighbourhoods of 0 for the topology 'Y. Let F 0 be the vecto.!... subspace of E, generated by the closures V in E (relative to 'Y ') of the sets V E 'B; the sets V form a fundamental system of neighbourhoods of 0 for a topology 'Yo on F 0' compatible with the vector space structure of F 0; for this topology, F 0 is complete, and the topology induced by 'Yo on'F is identical with 'Y.
6. Convex cones in topological vector spaces PROPOSITION 14. ~ In a topological vector space E, the closure ola convex set (resp. of a convex cone) is a convex set (resp. a convex cone with the same vertex). l)itmapsA x A in A, ~hic~ sho~s that ~ is c~nvex. Similarly, if C is a convex cone with vertex 0 then C + C c C and AC c C for all A > O. 4. ~ For any set A of a topological vector space E, the intersection of all the closed convex sets containing A is called the convex closed envelope of A; it is the smallest convex closed set containing A.
5) Let E and F be two complete metrisable vector spaces over a non-discrete valued division ring and let u be a linear mapping of E in F. Let N be the set of cluster points of u in F with respect to the filter of neighbourhoods of 0 in E; show that N is a closed vector subspace of F, and that, in order that u be continuous, it is necessary and sufficient that N be the single point 0 (use I, p. 19, cor. 5). Show that N is the smallest of the closed vector subspaces M of F such that, if denotes the canonical homomorphism of F on F 1M, then 0 u is a continuous mapping of E in F/M.
Algebraic Topology, Aarhus 1978: Proceedings of a Symposium held at Aarhus, Denmark, August 7-12, 1978 by J. L. Dupont, I. H. Madsen