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## Vector Space Concepts

Definition.A set of objects is called ametric spaceif 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 [Rudin 1964].

Definition.Alinear spaceis 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 , thenThe element is written as 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.

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- There exists in such that for all in .
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- .
- .
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Definition.Anormed linear spaceis a linear space on which there is defined a real-valued function of called anorm, denoted , satisfying the following three properties:The functional serves as a distance function on , so a normed linear space is also a metric space.

- , and .
- , a scalar.
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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.Apseudo-normis a real-valued function of satisfying the following three properties:A pseudo-norm differs from a norm in that the pseudo-norm can be zero for nonzero vectors (functions).

- , and .
- , a scalar.
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Definition.ABanach Spaceis acompletenormed linear space, that is, a normed linear space in which every Cauchy sequence^{2}*A sequence is said to be aCauchy sequenceif for each there is an such that for all and larger than . 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 [Royden 1968], p. 117.

Definition.A Hilbert space is a Banach space with a symmetric bilinearinner product defined such that the inner product of a vector with itself is the square of its norm .