Vol. 26, No. 1, 1968

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Dual groups of vector spaces

William Charles Waterhouse

Vol. 26 (1968), No. 1, 193–196

Let E be a topological vector space over a field K having a nontrivial absolute value. Let Ebe the dual space of continuous linear maps E K, and Ê the dual group of continuous characters E R∕Z. Ê is a vector space over K by ()(x) = φ(ax), and composition with a nonzero character of K is a linear map of Einto Ê. This map is always an isomorphism if K is locally compact, while if K is not locally compact it is never an isomorphism unless Ê = 0. When K is locally compact, Eis in addition topologically isomorphic to Ê 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 (Ê).

Mathematical Subject Classification
Primary: 46.01
Received: 20 December 1966
Published: 1 July 1968
William Charles Waterhouse