  After an appropriate restatement of the GNS construction for topological
$^*$-algebras we prove that there exists an isomorphism among the set
$\cycl(A)$ of weakly continuous strongly cyclic $^*$-representations of a
barreled dual-separable $^*$-algebra with unit $A$, the space $\hilb_A(A^*)$ of
the Hilbert spaces that are continuously embedded in $A^*$ and are
$^*$-invariant under the dual left regular action of $A$ and the set of the
corresponding reproducing kernels. We show that these isomorphisms are cone
morphisms and we prove many interesting results that follow from this fact. We
discuss how these results can be used to describe cyclic representations on
more general inner product spaces.
