A Hilbert space is a complete normed space whose norm is indexed by an inner (or scalar) product.
Two disjoint subspaces AA and BB of a space SS form a direct sum decomposition of SS if every element of SS can be written uniquely as a sum of an element of AA and an element of BB. The notation S=A⊕BS=A⊕B is then used.
A measurable function ff belongs to the Lebesgue space Lp(R),1≤p<∞Lp(R),1≤p<∞ if
f
p
=
∫

∞
+
∞
f
(
x
)
p
1
/
p
<
∞
.
f
p
=
∫

∞
+
∞
f
(
x
)
p
1
/
p
<
∞
.
(1)
An example of a Hilbert space is the Lebesgue space L2(R)L2(R) of measurable and square integrable functions. Indeed, the norm ·2·2 is induced by the scalar product
f
,
g
=
∫
f
(
x
)
g
(
x
)
¯
d
x
,
f
,
g
=
∫
f
(
x
)
g
(
x
)
¯
d
x
,
(2)
where g(x)¯g(x)¯ denotes the complex conjugate of g(x)g(x).
Two functions are said to be orthogonal in L2(R)L2(R) if their inner product is zero.
The Lebesgue measure can be replaced by a more general measure μμ, leading to the weighted space L2(μ)L2(μ), which has as inner product
f
,
g
d
μ
=
∫
f
(
x
)
g
(
x
)
¯
d
μ
(
x
)
f
,
g
d
μ
=
∫
f
(
x
)
g
(
x
)
¯
d
μ
(
x
)
(3)
and which contains the functions that have a finite norm fdμ:=f,fdμ<∞fdμ:=f,fdμ<∞.
A countable subset {fk}{fk} of functions belonging to a Hilbert space is a Riesz basis if every element ff of the space can be written uniquely as
f=∑kckfkf=∑kckfk, and if positive constants AA and BB exist such that
A
f
2
2
≤
∑
k
c
k
2
≤
B
f
2
2
.
A
f
2
2
≤
∑
k
c
k
2
≤
B
f
2
2
.
(4)
A Riesz basis is an orthogonal basis if the fkfk are mutually orthogonal. In this case, A=B=1A=B=1.