The inner product of two vectors/signals is the same as
the
ℓ2
ℓ
2
inner product of their expansion coefficients.

Let
b
i
b
i
be an orthonormal basis for a Hilbert Space
H
H.
x∈H
x
H
,
y∈H
y
H
x=∑
i
α
i
b
i
x
i
α
i
b
i
y=∑
i
β
i
b
i
y
i
β
i
b
i
then
〈x,y〉
H
=∑
i
α
i
β
i
¯
x
y
H
i
α
i
β
i

Applying the Fourier Series, we can go from
ft
f
t
to
c
n
c
n
and
gt
g
t
to
d
n
d
n
∫0Tftgt¯d
t
=∑
n
=−∞∞
c
n
d
n
¯
t
0
T
f
t
g
t
n
c
n
d
n
inner product in time-domain = inner product of Fourier
coefficients.

x=∑
i
α
i
b
i
x
i
α
i
b
i
y=∑
j
β
j
b
j
y
j
β
j
b
j
〈x,y〉
H
=〈∑
i
α
i
b
i
,∑
j
β
j
b
j
〉=∑
i
α
i
〈(
b
i
,∑
j
β
j
b
j
)〉=∑
i
α
i
∑
j
β
j
¯〈(
b
i
,
b
j
)〉=∑
i
α
i
β
i
¯
x
y
H
i
α
i
b
i
j
β
j
b
j
i
α
i
b
i
j
β
j
b
j
i
α
i
j
β
j
b
i
b
j
i
α
i
β
i
by using inner product rules

〈
b
i
,
b
j
〉=0
b
i
b
j
0
when
i≠j
i
j
and
〈
b
i
,
b
j
〉=1
b
i
b
j
1
when
i=j
i
j

If Hilbert space H has a ONB, then inner products are
equivalent to inner products in
ℓ2
ℓ
2
.

All H with ONB are somehow equivalent to
ℓ2
ℓ
2
.

square-summable sequences
are important

Comments:"My introduction to signal processing course at Rice University."