Metric spaces

We will view signals as elements of certain mathematical spaces. The spaces have a common structure, so it will be useful to think of them in theabstract.

Metric spaces

Definition 1

In this course we will assume familiarity with a number of common set operations. In particular, for the sets $A=\{0,1\}$ , $B=\left\{1\right\}$ , $C=\left\{2\right\}$ , we have the operations of:

• Union $A\cup B=\{0,1\}$ , $B\cup C=\{1,2\}$
• Intersection $A\cap B=\left\{1\right\}$ , $B\cap C=\varnothing$
• Exclusion $A\setminus B=\left\{0\right\}$
• Complement $A^c=U\setminus A$ , $A^c=\left\{2\right\}$
• Cartesian Product $A^2=A\times A=\{(0,0),(0,1),(1,0),(1,1)\}$

In order to be useful a set must typically satisfy some additional structure. We begin by defining a notion of distance.

Definition 2

• M1. $d(x,y)=d(y,x)$ (symmetry)
• M2. $d(x,y)\ge 0$ (non-negative)
• M3. $d(x,y)=0$ iff $x=y$ (positive semi-definite)
• M4. $d(x,z)\le d(x,y)+d(y,z)$ (triangle inequality).