Metric Spaces Functional Analysis

METRIC SPACES

1. METRICSPACES

Definition 1.1:[Metric Space]

LetX be a none-empty set. A mappingd :X×X −→Ris said to be a metric space on xif it satisfies the following conditions

(1) d(x,y)≥0 ∀x,y∈X

(2) d(x,y) =0⇔x=y ∀x,y∈X

(3) d(x,y) =d(y,x), ∀x,y∈X(Symmetric)

(4) d(x,y)≤d(x,z) +d(z,y) ∀x,y,z∈X (Triangle Inequality)

Example 1:LetX be any non-empty set .Define a mappingd:X×X−→Rbyd(x,y) =





0, ifx=y;

1, ifx6=y.

Thendis a metric onX,and this metric is called discrete metric.

Example 2: Let X =Fⁿ(Cⁿ,Rⁿ), n ≥1. Let x= (x1,x2,· · ·,xn),y= (y1,y2,· · ·,yn)∈Fⁿ Define d :X×X −→R by d(x,y) =

s n k=1

∑

|x_k−y_k|²= q

|x₁−y₁|²+|x₂−y₂|²+· · ·+|x_n−y_n|². To see the Triangle Inequality, we will need Minkowski InequalityIfai,bi∈F i=1,2,· · ·,n, and 1<p<∞, then

p

sn i=1

∑

|a_i+b_i|^p≤ ^p s n

i=1

∑

|a_i|^p+ ^p s n

i=1

∑

|b_i|^p

Now, letx= (x1,x2,· · ·,xn),y= (y1,y2,· · ·,yn),z= (z1,z2,· · ·,zn)∈Fⁿ

d(x,y) = s n

k=1

∑

|x_k−y_k|² Add and Subtractzk.

= s n

k=1

∑

|(x_k−z_k) + (z_k−y_k)|² Use Minkowski Inequality .

≤ s n

k=1

∑

|x_k−z_k|²+ s n

k=1

∑

|z_k−y_k|².

=d(x,z) +d(z,y)

November 15, 2008 1 © Dr.Hamed Al-Sulami

b

x r

FIGURE1

Example 3: lp.Letpbe a real number such that 1≤p<∞.lpis the space of all sequencex={xn}^∞_n=1inFsuch that

∑∞

n=1|x_n|^p<∞ (x={x_n}^∞_n=1converges).

l_p={x={x_n}^∞_n=1| ∑^∞

n=1|x_n|^p<∞, x_n,∈F,∀n∈N}

Define the mappingd:lp×lp−→Rbyd(x,y) =d({xn},{yn}) = ^p s ∞

n=1

∑

|xn−yn|^p.Thendis a metric onlp.

Example 4: C([a,b]).Leta,b be two real numbers such thata<b.C([a,b])is the space of all continuous real-valued functions f over[a,b].

C([a,b]) ={f :[a,b]→R| f is continuous on[a,b]}Define the mappingsd₁,d_∞:C([a,b])×C([a,b])−→Ras follows:

d_∞(f,g) = sup

x∈[a,b]

|f(x)−g(x)|and

d1(f,g) = Zb

a

|f(x)−g(x)|dx.Thend1,d∞are metrics onC([a,b]).

Open sets and closed sets.

Definition 1.2:[Basic Definition]

Let(X,d)be a metric space. LetE⊆X,and letx0∈X.

(1) Letx∈X andr>0.We define theopen ballof radiusraboutxto be the setB_r(x) ={y∈X|d(x,y)<r}.

(2) We say thatEis open set if for eachx∈Ethere is anε>0 such thatB_ε(x)⊆E. (3) We say thatEis closed set ifE^c=X\Eis open set.

(4) We say thatx0is an interior point ofEif there existr>0 such thatBr(x0)⊆E.

(5) We say thatx0is a limit point ofEif for eachr>0 ,Br(x0)∩(E\ {x0})6=φ.

(6) The set of all interior points ofEis denoted byE^◦. (7) The set of all limit points ofEis denoted byE⁰. (8) The closure set ofE,denoted byE, isE=E∪E⁰.

(9) We say thatx₀is a boundary point ofEif for eachr>0 ,B_r(x₀)∩E6=φandB_r(x₀)∩E^c6=φ.

(10) The set of all boundary points ofEis denoted by∂E.

Below you will find some of elementary results about metric space- you should try to prove them- Result 1.1:

Let(X,d)be a metric space. LetE⊆X.Then (1) φandXare both open and closed.

(2) An arbitrary union of open sets inXand a finite intersection of open sets inXare open sets inX.

(3) An arbitrary intersection of closed sets inXand a finite union of closed sets inX are closed sets inX. (4) Gis open⇔G=G^◦.

(5) Gis closed⇔G=G.

Definition 1.3:[Distance between sets and diameter ] Let(X,d)be a metric space. LetF,E⊆X,and letx0∈X.

(1) The distance betweenEandx0∈X is denoted byD(x0,E),is defined byD(x0,E) =inf

y∈Ed(x0,y).

(2) The distance between the setsEandF,denoted byD(F,E),is defined asD(F,E) = inf

x∈F,y∈Ed(x,y).

(3) The diameter of the setF, denoted byδ(F), is defined asδ(F) = sup

x,y∈Fd(x,y).

Result 1.2:

Let(X,d)be a metric space. LetE,F⊆X.Then (1) D(F,E) =D(E,F).

(2) Ifx∈E⇔D(x,E) =0.

(3) δ(E) =δ(E).

(4) IfE⊆F⇒δ(E)≤δ(F).

(5) E={x} ⇔δ(E) =0.

(6) Letx,y∈X,then|D(x,E)−D(y,F)| ≤d(x,y).

Convergence and completeness.

Definition 1.4:[Convergent and Cauchy Sequence]

Let(X,d)be a metric space. Let{xn} ⊆X,be a sequence.

(1) We say that{xn}is convergent tox∈Xif for eachε>0 there existN∈Nsuch that ifn>N⇒d(xn,x)<ε,and we write lim

n→∞xn=x.

(2) We say that{xn}is Cauchy if for eachε>0 there existN∈Nsuch that ifn,m>N⇒d(xn,xm)<ε.

(3) We say that{x_n}is bounded if there existM>0 such thatδ({x_n})≤M.

November 15, 2008 3 © Dr.Hamed Al-Sulami

Result 1.3:

Let(X,d)be a metric space. Let{x_n} ⊆X,be a sequence. Then

(1) If{xn}is convergent, then{xn}is bounded and its limit is unique.

(2) A Cauchy sequence is bounded.

(3) A convergent sequence is Cauchy.

(4) Ifxn→x,yn→yinX⇒d(xn,yn)→d(x,y).

(5) If{x_n}is convergent tox∈X, then every subsequence{x_nk}is convergent tox∈X.

(6) If{x_n}is Cauchy sequence and if{x_nk}is convergent subsequence tox∈X,then lim

n→∞x_n=x.

Definition 1.5:[Complete Metric]

Let(X,d)be a metric space. We say thatXis complete metric space if every Cauchy sequence inX converges to a point inX.

Example 5:

(1) The spacesRⁿ,Cⁿwithd(x,y) = r n

k=1∑|xk−yk|²are complete.

(2) The spacelpwith the metricd(x,y) =d({xn},{yn}) = ^p s ∞

n=1

∑

|xn−yn|^pis complete.

(3) The spacel_∞with the metricd(x,y) =d({x_n},{y_n}) =sup

n |x_n−y_n|is complete.

(4) The spaceC([a,b])with the metricd∞(f,g) = sup

x∈[a,b]

|f(x)−g(x)|is complete.

Example 6:

(1) The spacesQwithd(x,y) =|x−y|is not complete. For example the sequencex₁=1,x_n+1=x²_n+2

2xn is inQ, but

n→∞limx_n=√ 2∈/Q.

(2) The spaceC([−1,1])with the metricd1(f,g) = Z1

−1

|f(x)−g(x)|dx.is not complete. For example let

fn(x) =











1, if−1≤x≤0;

1−nx, if 0<x<1/n;

0, if 1/n≤x≤1.

. Then{fn}is Cauchy but it convergesχ_[−1,0]∈/C([−1,1])

Theorem 1.1: []

LetEbe a subset of a metric space(X,d)and letx∈X.Then (1) x∈E⇔ ∃{xn} ⊂E3 lim

n→∞xn=x.

(2) x∈E0 ⇒ ∃{xn} ⊂E3 lim

n→∞xn=x.

Proof:

(1) (⇒) Suppose thatx∈E. For eachn∈N,B_1/n(x)∩E6=φ.Pickx_n∈B_1/n(x)∩E.Now,d(x_n,x)<¹_n⇒lim

n→∞x_n=x.

(⇐) If{x_n} ⊂E3 lim

n→∞x_n=x,then for anyr>0, ∃ N∈N3ifn>N⇒d(x,x_n)<r.ThusB_r(x)∩E6=φ.

Hencex∈E.

(2) Suppose thst x∈E0. Choosex₁∈E such that x₁6=xand d(x,x₁)<1. Now, choose x₂∈E\ {x} such that d(x,x₂)<min{d(x,x₁),1/2}.Now, for eachn≥3 pickx_n∈E\ {x}such thatd(x,x_n)<min{d(x,x_n−1),1/n}.

Hence we have a sequence{xn} ⊂Esuch that lim

n→∞xn=x.

„

Dense sets and Separable Spaces.

Definition 1.6:[Separable Spaces]

Let(X,d)be a metric space. LetE⊆X. (1) We say thatEis dense ifE=X.

(2) We say thatX is separable if it has a countable dense subset.

Example 7:

(1) The spacesRⁿ,Cⁿwithd(x,y) = r _n

k=1∑|x_k−y_k|²are separable.

(2) The spacel_pwith the metricd(x,y) =d({x_n},{y_n}) = ^p s ∞

n=1

∑

|x_n−y_n|^p. (3) The spacel_∞with the metricd(x,y) =d({x_n},{y_n}) =sup

n |x_n−y_n|is not separable.

(4) The spaceC([a,b])with the metricd_∞(f,g) = sup

x∈[a,b]

|f(x)−g(x)|is separable.

Continuous Functions.

Definition 1.7:[Continuity]

Let(X,d₁),(Y,d₁)be a metric spaces. A function f:X→Yis called continuous atx₀∈Xif for eachε>0, ∃ aδ>0 such thatx∈Xandd₁(x,x₀)<δ⇒d₂(f(x),f(x₀))<ε. f is continuous onXif it is continuous at each point ofX. Theorem 1.2: []

Let(X,d₁),(Y,d₁)be a metric spaces and let f :X→Y be a function Then, the following statements are equivalent:

(1) f is continuous onX.

(2) ∀x∈Xand∀ {x_n} ⊂X3lim

n→∞x_n=x⇒ lim

n→∞f(x_n) =f(x).

(3) For eachEis open inY⇒ f⁻¹(E)is open inX (4) For eachFis closed inY ⇒f⁻¹(E)is closed inX (5) f(E)⊂f(E),∀ E⊆X.

(6) f⁻¹(F)⊂f⁻¹(F),∀ F⊆Y.

November 15, 2008 5 © Dr.Hamed Al-Sulami

Definition 1.8:[Homeomorphism]

Let(X,d₁),(Y,d₁)be a metric spaces.

(1) A function f:X→Y is called homeomorphism if f is a continuous bijective and f⁻¹is continuous.

(2) A function f:X→Y is called isometry ifd₂(f(x),f(y)) =d₁(x,y), x,y∈X.