**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

- .
- .
- There exists in such that for all in .
- .
- .
- .
- .

**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:

- , and .
- , a scalar.
- .

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:

- , and .
- , a scalar.
- .

**Definition. **A *Banach Space* is a *complete* normed linear
space, that is, a normed linear space in which every Cauchy
sequence^{1}converges 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
.

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