14. Vector Space Concepts

GUIDE: Elementary Digital Filter Theory. Vector Space Concepts

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Vector Space Concepts

Definition. A set $X$ of objects is called a metric spaceif with any two points $p$ and $q$ of $X$ there is associated a real number$d(p,q)$, called the distance from $p$ to $q$, such that (a) $d(p,q)>0$ if $p\; $d(p,p)=0$, (b) $d(p,q)=d(q,p)$, (c) $d(p,q)\, for any $r\ [Rudin 1964].

Definition. A linear space is a set of ``vectors'' $X$ together with a field of ``scalars'' ${\ with an addition operation $+:X\, and a multiplication opration $\taking ${\, with the following properties: If $x$, $y$, and $z$ are in $X$, and $\ are in ${\, then

  1. $x+y=y+x$.
  2. $x+(y+z)=(x+y)+z$.
  3. There exists $\ in $X$ such that $0\for all $x$ in $X$.
  4. $\.
  5. $(\.
  6. $1\.
  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 $X$ on which there is defined a real-valued function of $x\ called a norm, denoted $ \, satisfying the following three properties:

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

Note that when $X$ 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 $1$ 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 $x\ satisfying the following three properties:

  1. $\, and $x=0\.
  2. $\, $c$ 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 sequence2*A sequence $H_n(e^{j\ is said to be a Cauchy sequence if for each $\ there is an $N$ such that $ \ for all $n$ and $m$ larger than $N$. converges to an element of the space.

Definition. A function $H(e^{j\ is said to belong to the space $Lp$ if

\

Definition. A function $H(e^{j\ is said to belong to the space $H^p$ if it is in $Lp$ and if its analytic continuation $H(z)$ is analytic for$\. $H(z)$ is said to be in $H^{-p}$ if $H(z^{-1})\.

Theorem. (Riesz-Fischer) The $Lp$ spaces are complete.Proof. See Royden [Royden 1968], p. 117.

Definition. A Hilbert space is a Banach space with a symmetric bilinearinner product $<x,y>$ defined such that the inner product of a vector with itself is the square of its norm $<x,x>= \.

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