# Connexions

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# Metric Spaces

Module by: Mark A. Davenport. E-mail the author

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 the abstract.

## Metric Spaces

### Definition 1

A set is a (possibly infinite) collection of distinct objects.

### Example 1

• The empty set: ={}={} (plays a role akin to zero)
• Binary numbers: {0,1}{0,1}
• Natural numbers: N={1,2,3,...}N={1,2,3,...}
• Integers: Z={...,-2,-1,0,1,2,...}Z={...,-2,-1,0,1,2,...} (ZZ is short for “Zahlen”, German for “numbers”)
• Rational numbers: QQ (QQ for “quotient”)
• Real numbers: RR
• Complex numbers: CC

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

• Union: AB={0,1}AB={0,1}, BC={1,2}BC={1,2}
• Intersection: AB={1}AB={1}, BC=BC=
• Exclusion: A B = { 0 } A B = { 0 }
• Complement: Ac=UAAc=UA, Ac={2}Ac={2}
• Cartesian Product: A 2 = A × A = { ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) } A 2 = A × 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

A metric space is a set MM together with a metric (distance function) d:M×MRd:M×MR such that for all x,y,zMx,y,zM

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

### Example 2

• Trivial metric: (MM is arbitrary) d(x,y)=0 if x=y,1 if xy.d(x,y)=0 if x=y,1 if xy.
• Standard metric: (M=RM=R) d(x,y)=|x-y|d(x,y)=|x-y|
• Euclidean (2)(2) metric: (M=RNM=RN) d(x,y)=i=1N|xi-yi|2d(x,y)=i=1N|xi-yi|2
• 11 metric: (M=RNM=RN) d(x,y)=i=1N|xi-yi|d(x,y)=i=1N|xi-yi|
• pp metric, 1p<1p<: (M=RNM=RN) d(x,y)=i=1N|xi-yi|p1/pd(x,y)=i=1N|xi-yi|p1/p
• metric: (M=RNM=RN) d(x,y)=maxi=1,...,N|xi-yi|d(x,y)=maxi=1,...,N|xi-yi|
• LpLp metric: (MM = real (or complex) valued functions defined on [a,b][a,b]) dp(x,y)=ab|x(t)-y(t)|pdt1/pdp(x,y)=ab|x(t)-y(t)|pdt1/p

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