Is the infinite complex projective space related to a projective separable Hilbert space?

Question

For $H$ a Hilbert space, let $P(H)$ denote its projective Hilbert space. Let $\mathbb CP^\infty$ be the infinite complex projective space (eg constructed inductively as a CW complex). Note that $\mathbb CP^{d-1} = P(\mathbb C^d)$. Analogously, if $H$ is a separable Hilbert space, eg $\ell^2(\mathbb N)$, is $P(H)$ related to $\mathbb CP^\infty$?