I was pondering this question in class earlier:
All separable, infinite dimensional Hilbert spaces are isometrically isomorphic. Thus, in particular, any such space is isometrically isomorphic to $L^2[a,b]$, for example. The latter is a rather concrete space of (equivalence classes of) functions. The question is, can you think of an interesting Hilbert space that don't immediately look like a function space, but because of this isomorphism, can be thought of as one?
Hope this makes sense. Not an important question, just a curiosity.
The space of Hilbert–Schmidt operators is a Hilbert space with the inner product $\langle A,B\rangle = \operatorname{tr}(A^*B)$.
And I suppose that sequences are considered as a special kind of functions. Then... I don't know of any other source of Hilbert spaces.