Metric Spaces

Adrian Down 16779577

August 2, 2005

1 Metrics: a notion of distance

So far in analysis on R, every deﬁnition (convergence, Cauchy, continuity)

has included statements like “|x−y|< ”. The idea is that this |x−y|

represents a distance from xto y.

A distance should satisfy the properties:

i) |x−y| ≥ 0,∀x, y ∈R

ii) |x−y|= 0 →x=y

iii) |x−y|=|y−x|

iv) “Triangle inequality”: |x−y|+|y−z| ≥ |x−z|,∀x, y, z

Deﬁnition: a metric space is a set of “objects” Stogether with a function

don the set of pairs (x, y), x, y ∈Ssuch that

d(x, y)∈R,∀x, y ∈S

1)d(x, y)≥0,∀x, y ∈S

2)d(x, y) = 0 →x=y

3)d(x, y) = d(y, x),∀x, y ∈S

4)d(x, y) + d(y, z)≥d(x, z),∀x, y, z ∈S

dis called the metric on S. May write (S, d) to specify the metric.

4) is called the triangle inequality; it is the most important, and it is

usually the hardest to verify.

2 Examples

1) (R, d) is a metric space where d(x, y) = |x−y|

1

2) “Discrete metric space”: Sis any set, and deﬁne

d(x, y) = (1x, y ∈S, x 6=y

0x∈S, x =y

3) “Euclidean k-space”:

S={(x1, x2, . . . , xk)|xi∈R,1≤i≤k}

d(x,y) = k

X

i=1

(xi−yi)2!

1

2

Properties 1, 2, and 3 are clear. Property 4 takes work. Best way is to

consider dot products of vectors.

This is the most important metric space that we will cover. The given

metric space is called the standard metric on Rk.

4) “Manhattan metric”:

S=Rk

d(x,y) =

k

X

i=1 |xi−yi|

Properties 1, 2, and 3 are easy. For 4, use the triangle inequality on R,

X|xi−yi|+X|yi−zi| ≥ X|xi−zi|

5) C[0,1] :

S={f|fis a continous function on [0,1]}

d(f, g) = sup{|f(x)−g(x)||x∈[0,1]}

Note that f, g continuous on [0,1] →f−gis continuous on [0,1] →d(f, g)

exists.

To verify property 2,

d(f, g) = 0 →sup{|f(x)−g(x)|} = 0 → |f(x)−g(x)|= 0,∀x→f(x) = g(x),∀x

2

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Subject:
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08/10/2011