TITLE: Fubini-Study metric for an infinite dimensional Hilbert space
QUESTION [5 upvotes]: Hi,
Can one define a Fubini-Study metric/Kaehler metric on the projective space of an infinite dimensional Hilbert space, i.e. using the formula $\partial \bar{\partial} \log |Z|^2$? 
This should be very well-known to the experts. Anyhow I don't have much experience with infinite dimension and worried that something may go wrong. I appreciate any comments or references.

REPLY [3 votes]: Maybe it's easier to see that the definition extends if you don't use a formula:
The unit sphere inherits a Riemannian metric from the Hilbert space in the standard manner and since it is invariant under the circular symmetry $(e^{i\theta},x) \mapsto  e^{i\theta} x$, it will
project down to a Riemannian metric on the complex projective space.