(Basis in Hilbert space) The countable
collection of elements
is a "basis" in Hilbert space
if any element of
may be approximated with arbitrary precision by a finite linear combination of
elements from
.
A Hilbert space is called "separable" if it has a basis.

Proposition

The countable collection
of elements
is a basis in Hilbert space
iff

Definition

(Fourier decomposition) The "Fourier
decomposition" of
with respect to the basis
is the
series
The quantities
are called "Fourier coefficients".

Proposition

(Main property of Fourier
decomposition) For a family
with the property
and any
the
minimum
is attained at
.
If
is a basis
then

Proposition

(Bessel equality) For a family
with the property
and any
,

Proposition

(Parseval equality) Assume that
is a basis in
.
For any
we
have