Definition.
A set of objects is called a metric space if with any two
points
and
of
there is associated a real number
,
called the distance from
to
, such that (a)
if
;
, (b)
, (c)
, for any
[6].
Definition. A linear space is a set of ``vectors'' together
with a field of ``scalars''
with an addition
operation
, and a multiplication opration
taking
, with the following properties: If
,
,
and
are in
, and
are in
, then
Definition.
A normed linear space is a linear space on which there is defined
a real-valued function of
called
a norm, denoted
,
satisfying the following three properties:
Note that when is the space of continuous complex functions
on the unit circle in the complex plane, the norm of a function is not
changed when multiplied by a function of modulus
on the unit circle.
In signal processing terms, the norm is insensitive to multiplication by a
unity-gain allpass filter (also known as a Blaschke product).
Definition.
A pseudo-norm is
a real-valued function of
satisfying the following three properties:
Definition. A Banach Space is a complete normed linear
space, that is, a normed linear space in which every Cauchy
sequence1converges to an element of the space.
Definition. A function
is said to belong to the space
if
Definition. A function
is said to belong to the space
if
it is in
and if its analytic continuation
is analytic for
.
is said to be in
if
.
Theorem. (Riesz-Fischer) The spaces are complete.
Proof. See Royden [5], p. 117.
Definition. A Hilbert space is a Banach space with a symmetric bilinear
inner product defined such that the inner product of a vector with
itself is the square of its norm
.