Digital Signal Processing Reference
In-Depth Information
pleteness. This is a slightly more technical notion, which essentially im-
plies that convergent sequences of vectors in
V
have a limit that is also in
V
. To gain intuition, think of the set of rational numbers
versus the set of
real numbers . The set of rational numbers is incomplete, because there
are convergent sequences in
l
g
r
,
y
i
d
.
,
©
,
L
s
which converge to irrational numbers. The
set of real numbers contains these irrational numbers, and is in that sense
the completion of
. Completeness is usually hard to prove in the case of
infinite-dimensional spaces; in the following it will be tacitly assumed and
the interested reader can easily find the relevant proofs in advanced analysis
textbooks. Finally, we will also only consider
separate
Hilbert spaces, which
are the ones that admit orthonormal bases.
3.2.2
Examples of Hilbert Spaces
Finite Euclidean Spaces.
The vector space
N
, with the “natural” defi-
nition for the sum of two vectors
z
=
x
+
y
as
z
n
=
x
n
+
y
n
(3.28)
and the definition of the inner product as
N
−
1
x
∗
n
y
n
〈
x
,
y
〉
=
(3.29)
n
=
0
is a Hilbert space.
Polynomial Functions.
An example of “functional” Hilbert space is the
vector space
N
[
]
of polynomial functions on the interval
[
]
0, 1
0, 1
with
∞
[
]
is not com-
maximum degree
N
. It is a good exercise to show that
0, 1
plete; consider for instance the sequence of polynomials
n
x
k
k
!
p
n
(
x
)=
k
=
0
∈
∞
[
]
.
e
x
This series converges as
p
n
(
x
)
→
0, 1
Square Summable Functions.
Another interesting example of func-
tional Hilbert space is the space of
square integrable functions
over a finite
interval. For instance,
L
2
[
−
π
π
]
is the space of real or complex functions
,
on the interval
[
−
π
,
π
]
whichhaveafinitenorm. Theinnerproductover
L
2
[
−
π
,
π
]
is defined as
π
f
∗
(
〈
f
,
g
〉
=
t
)
g
(
t
)
dt
(3.30)
−
π
