Maybe I am wrong, but the Wikipedia account looks a little bit suspect to me: they describe a rigged Hilbert space as consisting of a pair of inclusions $i: S \to H$, $j: H \to S^\ast$ of topological vector space inclusions, where $S^\ast$ is the strong dual of $S$, $H$ is a (separable) Hilbert space, $i$ is dense, and $j$ is the conjugate linear isomorphism $H \simeq H^\ast$ followed by the adjoint $i^\ast: H^\ast \to S^\ast$. This seems a little vague to me; should $S$ be more specifically a nuclear space or something? My guess is that a typical application would be where $S$ is Schwartz space on $\mathbb{R}^4$, with its standard dense inclusion in $L^2(\mathbb{R}^4)$, so $S^\ast$ consists of tempered distributions.