I am asking this because I'm trying to wrap my head around rigged Hilbert spaces (see https://en.wikipedia.org/wiki/Rigged_Hilbert_space#Formal_definition_(Gelfand_triple) ). In RHS theory, it is usually talked about a triplet ($\phi$, $H$, $\phi^*$), where $H$ is an infinite dimentional Hilbert space, $\phi\subset H$ is dense and $\phi^*$ is its dual. In this context, one has the following relation:
$$\phi\subset H=H^*\subset \phi^*\,,$$
which implies that $\phi$ is not itself a Hilbert space as a Hilbert space and its dual should de isomorphic (and since it can be associated to the same scalar product as H, it would further mean that $\phi$ is actually not complete).
My question would be: is this always the case (i.e. any dense subset of a Hilbert space is not complete) or it just happens in certain cases?
My atempt at explaining why it is always the case is: Since all infinite dimensional Hilbert spaces are isomorphic to an $l^2$ space, $\phi$ being a Hilbert space implies that $\phi$ is isomorpric to H, which contadicts the proper subset statement. Does this make sense?
