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 (not in) 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? 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.