International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 47

The space of (equivalence classes of) squareintegrable complexvalued functions f on the circle is contained in , since by the Cauchy–Schwarz inequalityThus all the results derived for hold for , a great simplification over the situation in or where neither nor was contained in the other.
However, more can be proved in , because is a Hilbert space (Section 1.3.2.2.4) for the inner productand because the family of functions constitutes an orthonormal Hilbert basis for .
The sequence of Fourier coefficients of belongs to the space of squaresummable sequences:Conversely, every element of is the sequence of Fourier coefficients of a unique function in . The inner productmakes into a Hilbert space, and the map from to established by the Fourier transformation is an isometry (Parseval/Plancherel):or equivalently:This is a useful property in applications, since (f, g) may be calculated either from f and g themselves, or from their Fourier coefficients and (see Section 1.3.4.4.6) for crystallographic applications).
By virtue of the orthogonality of the basis , the partial sum is the best meansquare fit to f in the linear subspace of spanned by , and hence (Bessel's inequality)