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## Vector Space Concepts

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

1. .
2. .
3. There exists in such that for all in .
4. .
5. .
6. .
7. .
The element is written as 0 thus coinciding with the notation for the real number zero. A linear space is sometimes be called a linear vector space, or a vector space.

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:

1. , and .
2. , a scalar.
3. .
The functional serves as a distance function on , so a normed linear space is also a metric space.

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:

1. , and .
2. , a scalar.
3. .
A pseudo-norm differs from a norm in that the pseudo-norm can be zero for nonzero vectors (functions).

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 .

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