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.



Fubini-Study metric for an infinite dimensional Hilbert space
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. 


