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 .
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 , 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 .