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Some of the answers to this question might be helpful for your question also. It deals with finite-dimensional Hilbert spaces, but most of my answer to that question applies to the infinite-dimensional case too, with one or two obvious exceptions (e.g. the metric space of 1-dimensional subspaces of an infinite-dimensional Hilbert space is not compact). In particular, the book on Hilbert spaces by Akhiezer and Glazman has a short (5 pages?) section on the Grassmannian of a Hilbert space, and shows that the metric on the Grassmannian given by `aperture' is the same as the metric given by the operator difference between orthogonal projections.