# Fubini-Study metric for an infinite dimensional Hilbert space

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.

-

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.

-
Thanks Ahmed and Alvarez, so definition works as in the finite dimensional case. –  Kiu Apr 2 '13 at 3:20