Continuity of representation of topological group

Solution 1:

For spaces $X,Y$ let $C(X,Y)$ denote the set of continuous functions $X \to Y$. This set endowed with the compact-open will be denoted by $Y^X$. There is a canonical function $E : C(X \times Y,Z) \to C(X,Z^Y)$ where for $f : X \times Y \to Z$ we define $E(f) : X \to Z^Y$ by $E(f)(x) : Y \to Z, E(f)(x)(y) = f(x,y)$. This function is known as the exponential correspondence. See any book on general topology treating function spaces. There are also a number of contributions in this forum, for example When is the exponential law in topology discontinuous? (search for "exponential law").

The function $E$ is trivially injective, but surjectivity requires to assume that $Y$ is locally compact.

For your question this means that (2) implies (1). The converse is true under the additional assumption that $V$ is locally compact. Thus, if $k$ is a locally compact topological field, then (1) and (2) are equivalent because $V \approx k^n$ is locally compact.