Dice $ x_1, ldots, x_n in H $ in a Hilbert space, the volume of a parallelotope can be calculated as the square root of the Gramm determinant:

$$

V (x_1, ldots, x_n) = sqrt {G (x_1, ldots, x_n)},

$$

where

$$

G (x_1, ldots, x_n) = det langle x_i, x_j rangle_ {i, j = 1} ^ n.

$$

While it has a perfect sense in the real Hilbert space, you can also use it to define the volume in the complex.

To see a test and some amazing applications (in real Hilbert space), see Sections 5.3 and 5.3 in:

http://www.pitt.edu/~hajlasz/Notatki/Functional%20Analysis2.pdf.