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 Ê the dual group of
continuous characters E → R∕Z. Ê is a vector space over K by (aφ)(x) = φ(ax),
and composition with a nonzero character of K is a linear map of E′ into
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, E′ is 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)∧.
