topological vector spaces
locally convex topological vector spaces
topological vector space
locally convex topological vector space
Hilbert Spaces, Fréchet Spaces, Sobolev spaces, Lebesgue Spaces
Bornological Vector Spaces
Barrelled Vector Spaces
bounded, unbounded, self-adjoint, compact, Fredholm
spectrum of an operator
Riesz representation theorem
Algebraic Theories in Functional Analysis
An Elementary Treatment of Hilbert Spaces
When are two Banach spaces isomorphic?
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The theorem of (Alfsen-Shultz 78) says that two C*-algebras have isomorphic state spaces precisely if as Jordan algebras they are isomorphic.
This is to some extent a Jordan algebra-analog of the Gelfand-Naimark theorem.
Other theorems on the foundations of quantum mechanics include
order-theoretic structure in quantum mechanics