  Hilbert space combines the properties of two fundamentally different types of
mathematical spaces: vector space and metric space. While the vector-space
aspects of Hilbert space, such as formation of linear combinations of state
vectors, are routinely used in quantum mechanics, the metric-space aspects of
Hilbert space are much less exploited. Here we show that a suitable metric
stratifies Fock space into concentric spheres. Maximum and minimum distances
between wave functions are derived and geometrically interpreted in terms of
this metric. Unlike the usual Hilbert-space analysis, our results apply also to
the reduced space of only ground-state wave functions and to that of particle
densities, each of which forms a metric space but not a Hilbert space. The
Hohenberg-Kohn mapping between densities and ground-state wave functions, which
is highly complex and nonlocal in coordinate description, is found, for three
different model systems, to be very simple in metric space, where it is
represented by a monotonic mapping of vicinities onto vicinities. Surprisingly,
it is also found to be nearly linear over a wide range of parameters.
