Being nonseparable Banach space is in fact nothing special: one meets the first
examples in the standard functional analysis course, when one learns about
$\ell^p$ or $L^p[0,1]$ spaces-these spaces are nonseparable when $p=\infty$.
However I spoke to one person which said to me that nonseparable Hilbert
spaces are rather exotic and in "real life" one rarely come across them. Here I list some examples when I came across nonseparable Hilbert spaces—but to be honest, none of them have truly convinced me about the importance of those spaces:
-
When $A$ is a $C^*$-algebra, then if $A$ is separable then $A$ embeds into $B(H)$ where $H$ is separable—nevertheless sometimes one is interested in so called universal representation $\pi_u$ of $A$ (one feature of this representation is that any state of $\pi_u(A)$ is in fact a vector state). The Hilbert space of this representation is hardly ever separable.
-
When $Q$ denotes the Calkin algebra (the quotient of $B(H)$ by compact operators) one can show that it cannot be represented on separable Hilbert space (this is very elegant argument: there is an uncountable family of infinite subsets of $\mathbb{N}$ such that the intersection of any two members of this family is finite—this gives an uncountable family of mutually disjoint projections in the quotient—in fact this argument applies also to $\ell^{\infty}/c_0$).
-
There is a notion of almost periodic function on $\mathbb{R}$ (this can be generalized to other groups but let us stick to this particular case) and the space of almost periodic functions contains all functions of the type $e_s(t)=e^{ist}$. One can introduce the scalar product on the linear span of all $e_s$-s and those function turns out to be mutually orthogonal. Completing one obtains a nonseparable Hilbert spaces.
-
There is a notion of tensor product of Hilbert spaces in particular the so called complete tensor product which is due to von Neumann. This construction yields a nonseparable Hilbert space when the tensored family is infinite. But as far as I know, in most application one restricts to the preferred so called $C_0$-sequence thus obtaining the separable space (this is some sort of choosing a ,,stabilization'').
That's all-I don't know any other examples of situations when one is faced with nonseparable Hilbert spaces-so ,,huge'' for the first sight spaces as Fock spaces used in second quantization or $L^2$ over some infinite dimensional spaces with Gaussian measures are all separable.
So I would like to ask:
What are some interesting examples of nonseparable Hilbert spaces which you have ever met?
Forgive me if my question is too vaque but I tried to give some motivation around it.
Best Answer
To the contrary, my feeling is that nonseparable Hilbert spaces are in some sense artifacts and can almost always be avoided. And more generally, to your comment about nonseparable Banach spaces being "nothing special", most of the nonseparable Banach spaces one meets in practice are duals of separable Banach spaces, and therefore are weak* separable. Algebras of almost periodic functions, and their noncommutative analogs, the CCR algebras, are rare examples of interesting nonseparable Banach spaces which are not dual spaces, but even here one is mainly interested in regular representations, which are determined by their behavior on separable subalgebras (the span of the functions $e_s$ with $s$ rational, say).