I know that Schwartz space can be considered a dense subset of the Hilbert space isomorphic to $\ell^2$. What I wish to understand is, how really different Schwartz space is from the Hilbert space.
Schwartz space has completeness, and is an inner product space, at least understood as "borrowing" inner product structure from the Hilbert space. This seems to satisfy all the definitions of the Hilbert space. Yet we know that these two spaces are not isomorphic. So what distinguish them?
Maybe the problem is that inner product borrowed from the Hilbert space is not natural to the natural topology of Schwartz space? It would be nice if someone clarifies on this.
And I believe quantum mechanics uses inner product on Schwartz space, so I guess Schwartz space is understood at least there as an inner product space...