What is known about this space of parametrised Hilbert spaces? For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega_s)$, where $\Omega_s$ is some nice domain that depends on $s$ in a nice way.
Define $H = \{H(s) : s \in [0,\infty)\}$ the set containing all Hilbert spaces $H(s)$.
What kind of space is $H$? Can we put a norm on it or a vector space structure? What is known about such spaces of Hilbert spaces? Thanks.
I ask this question because I wish to think of convergence of Hilbert spaces.
 A: Perhaps the most common type of convergence of metric spaces is Gromov-Hausdorff convergence. However, isometric spaces are distance 0 in the Gromov-Hausdorff metric, so this doesn't work well here.
Another idea is convergence of subsets. By extending functions by zero, you can make each of your Hilbert spaces a subset of $L^2(R^n)$. Then a sequence of subsets convereges if its lim sup is equal to its lim inf, and we all the resulting subset the limit.
This will probably be the same as just looking at limits of your domains.
A: You can obviously equip $H$ with a vector space structure, by letting 
$$(x_\alpha)_{\alpha\in [0, \infty)}+(y_\alpha)_{\alpha\in [0, \infty)}=(x_\alpha+y_\alpha)_{\alpha\in [0, \infty)}$$
and
$$\lambda(x_\alpha)_{\alpha\in [0, \infty)}=(\lambda x_\alpha)_{\alpha\in [0, \infty)},$$
where of course $x_\alpha, y_\alpha\in H(\alpha)$. However, you cannot define a Hilbert norm in some natural way on the whole of $H$, for the same reason why you cannot define a Hilbert norm on the whole space of functions $[0, \infty)\to \mathbb{R}$. What you can do is introduce a subspace 
$$\bigoplus_{s\in [0, \infty)} H(s)=\left\{ (x_\alpha)_{\alpha\in [0, \infty)}\ :\ \int_0^\infty \lVert x_\alpha\rVert^2\, d\alpha < \infty\right\}, $$
which you will equip with the norm 
$$\lVert (x_\alpha)\rVert=\sqrt{\int_0^\infty \lVert x_\alpha\rVert^2\, d\alpha}.$$
I have not checked but this will most surely be a complete space if each of the $H(s)$ is. 
