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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 $\mathbb CP^\infty$?

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  • $\begingroup$ They're at least weakly homotopy equivalent, and I think likely homotopy equivalent. $\endgroup$ Oct 11, 2022 at 18:45

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