Let $\mathcal{H}$ be an infinite dimensional separable (complex) Hilbert space. What is a natural space which parameterizes the choices of orthonormal bases for $\mathcal{H}$?


It seems like one option, in analogy to the finite-dimensional Stiefel manifold (something I know nothing about beyond what I just glanced at on the wikipedia page), would be to consider the set of all unitary operators from $\ell^2(\mathbb{N})\to\mathcal{H}$, with some topology. What kind of topology would one consider? Ideally, I'd like it to be Polish.



A space parameterizing the choices of orthonormal bases for a Hilbert space
You could look at the set of operators in $B(H)$ which are diagonalized by a given basis. For each choice of orthonormal basis, this gives you an atomic maximal abelian von Neumann algebra (atomic masa). Now there is a natural topology on the space of all von Neumann algebras sitting inside $B(H)$ called the Effros-Maréchal topology, so you could just restrict this topology to the set of atomic masas.


[Here](http://www.sciencedirect.com/science/article/pii/S0022123699935383) is a link that may be helpful.


