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|

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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

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To verify property 4, use the triangle inequality on R,

d(f, g) + d(g, h) = sup{|f(x)−g(x)|} + sup{|g(x)−h(x)|}

≥sup{|f(x)−g(x)|+|g(x)−h(x)|} ≥ sup{|f(x)−h(x)|} =d(f, h)

6) l1(N) :

S={(an)∞

n=1|

∞

X

n=1 |an|converges}

d((an),(bn)) =

∞

X

n=1 |an−bn|

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

d((an, bn)) + d((bn),(cn)) =

∞

X

n=1 |an−bn|+

∞

X

n=1 |bn−cn|

=

∞

X

n=1 |an−bn|+|bn−cn| ≥

∞

X

n=1 |an−cn|=d((an),(cn))

Idea is to abstract the qualities of Rto many situations. The multitude

of examples is one of the most useful aspects of metric spaces.

3 Convergence

Deﬁnition: A sequence (sn)∞

n=1 in a metric space (S, d) is said to converge to

s∈Sif

∀ > 0,∃N∈Nsuch that n>N→d(sn, s)<

Note: snconverges to sif and only if limn→∞ d(sn, s) = 0 in R.

Deﬁnition: a sequence (sn) is (S, d) is Cauchy if

∀ > 0,∃N∈Nsuch that n, m > N →d(sn, sm)<

Deﬁnition: A metric space (S, d) is said to be complete if every Cauchy

sequence converges.

Note: the converse statement that convergence implies Cauchy is always

true. Emulate the proof from R.

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Examples:

1) (R, d), d(x, y) = |x−y|is complete by the Completeness axiom

2) (Q, d), d(x, y) = |x−y|is not complete

3) C[0,1] is complete.

4) (Rk, d), dis the Euclidean metric, is complete. To prove it, we need

lemmas

Lemma 1: (xn) is a convergent sequence in Rk→(xn

i)∞

n=1 is convergent

in R,∀i, 1≤i≤k.

Proof: Assume (xn) is convergent and xnconverges to x. Let > 0 be

given. ∃Nsuch that

n>N→ k

X

i=1

(xn

i−xn

j)2!

1

2

< →X(xn

i−xi)2< 2

→(xn

i−xi)2< 2,∀i→ |xn

i−xi|< , ∀i→lim

n→∞

xn

i=xi,∀i

Now assume that (xn

1),(xn

2),...,(xn

k) converge in R. Let

lim

n→∞

xn

1=x1,..., lim

n→∞

xn

k=xk

Let > 0 be given. Then ∃N1, . . . , Nk∈Nsuch that

n>N1→ |x1−xn

1|<

√k, . . . , n > Nk→ |xk−xn

k|<

√k

Let N= max{N1, N2, . . . , Nk}. Then

Lemma 2: (xn) is Cauchy in Rkif and only if (xn

i) is Cauchy in R,∀i

Proof: left to the reader

Now, if (xn) is Cauchy in Rk(xn

i) is Cauchy in R,∀i→(xn

i) is convergent

∀i→(xn) is convergent in Rk.

So Rkis complete.

What’s useful are the lemmas: to examine the properties of something in

Rk, look at the components of the vector one at a time.

Deﬁnition: A set A⊆S, (S, d) is a metric space is bounded if ∃x∈Sand

∃M∈Rsuch that d(a, x)≤M, ∀a∈A.

Deﬁnition: the open ball around xwith radius Ris BR(x) = {s∈

S|d(s, x)< R}.

The closed ball =¯

BR(x) = {s∈S|d(s, x)≤R}.

Ais bounded if A⊆¯

BR(x) for some x, R.

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