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? Assume whatever is necessary for $\Omega\_s$ (eg. $\Omega\_0 \subset \Omega\_s$ for all $s > 0$).


I ask this question because I wish to think of convergence of (not *in*) Hilbert spaces.



Parametrised Hilbert spaces; can we put a norm on the following space of Hilbert spaces?
I think that the concept you are looking for is that of a measurable field of Hilbert spaces. These are studied in detail in Dixmier's classic "von Neumann algebras" which is available on Google Books. In particular he shows how to define
a Hilbert space from such a field, the latter being exactly as in your query a family of Hilbert spaces depending in a suitable manner on a parameter.


