linear algebra: volume of paralelotope in $ L ^ 2 ( mathbb R). $

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.