Dual groups of vector spaces

Abstract

Let E be a topological vector space over a field K having a nontrivial absolute value. Let E' be the dual space of continuous linear maps E->K, and E the dual group of continuous characters E →R/Z. E is a vector space over K by (a→)(x) = →(ax), and composition with a nonzero character of K is a linear map of E' into E. This map is always an isomorphism if K is locally compact, while if K is not locally compact it is never an isomorphism unless E = 0. When K is locally compact, E' is in addition topologically isomorphic to E if each is given its topology of uniform convergence on compact sets. This leads to conditions on E which imply that E is topologically isomorphic to (E). © 1968 by Pacific Journal of Mathematics.