Same scheme can be applied to functions instead of numbers.

- Consider a function
- Function is transformed to the series of its Fourier coefficients

by the Fourier transform:

The sequence is bounded by the inequality

for any function .

- We may complement of the type (1) to the set of all sequences, with may have unconstrained sum of the squares of their elements.
- Applying then the inverse Fourier transform we define the expansion of the
functional space to the space of
*distributions*.