Euclidean Space and Metric Spaces - UCI Mathematics

1 Chapter 8 Euclidean Space and Structures on Euclidean Vector and Metric SpacesThe setKnofn-tuplesx= (x1, x2.. , xn) can be made into avector spaceby introducing the standard operations of addition and scalar multiplicationthroughx+y= (x1, x2.. , xn) + (y1, y2.. , yn) := (x1+y1, x2+y2.. , xn+yn), x= (x1, x2.. , xn) := ( x1, x2.. , xn), K, x, y for the topology ofKnwe introduce the distance functiond(x, y) :=[n k=1|xk yk|2]1/2which satisfies the following properties(m1)d(x, y) 0 x, y Knandd(x, y) = 0 x=y.(m2)d(x, y) =d(y, x) x, y Kn.(m3)d(x, z) d(x, y) +d(y, z) x, y, z Kn(triangle-inequality)Definition pair (M, d) is calledmetric spaceiff(i)Mis a set.

2 (ii)d:M M [0, ) satisfies (m1)-(m3).125126 CHAPTER 8. Euclidean Space AND Metric SPACESE xamples (a)(Kn,[ nk=1|xk yk|2]1/2)is a Metric Space .(b) (R, d0) is a Metric Space ford0(x, y) :={0, x=y1, x6=yx, y R(c) IfM=Knanddpis defined bydp(x, y) :={[ nk=1|xk yk|p]1/p,1 p < ,maxk=1,..,n|xk yk|, p= ,then (M, dp) is a Metric Spaces forp [1, ]. Norms and Scalar ProductsObserve thatd2(x, y) only depends on thex y. In particular, by defining|x|2:= nk=1x2k, we recoverd(x, y) =|x y| pair (V,| |V) is anormed vector spaceif(i)Vis aK-vector Space .(ii)| |V:V [0, ) is a norm onV, that is, it satisfies(n1)|x|V 0 x V,|x|= 0 x= 0.]]}}

3 (n2)| x|V=| ||x|V x V K.(n3)|x+y| |x|+|y| x, y V. (triangle inequality)Remarks (a) If (V,| |V) is a normed vector Space , then (V, dV) is ametric Space fordv(x, y) :=|x y|V x, y V .(b) For eachp [1, ]|x|p:=(n k=1|xk yk|p)1/pis a norm onV=Rn,Cn.(c) IfV= C([a, b])3fand f := supx [a,b]|f(x)|then (V, ) is anormed vector Space .(d) IfV= C1([a, b])3fand f 1, := f + f then (V, 1, ) isa normed vector STRUCTURES ON Euclidean SPACE127 Definition a vector Space . Then , =V V K,(x, y)7 x, y is calledinner productif(i1) x, y = y, x x, y V.(i2) x+ y, z = x, z + y, z and x, y+ z = x, y + x, z x, y, x V K.

4 (i3) x, x 0 x Vand x, x = 0 iffx= 0. The pair(V, , )is calledinner product C([a, b],K) and f, g := baf g dxforf, g V, then(V, , )is an inner product (Cauchy-Schwarz)Let(V, , )be an inner product Space . Then| x, y | x, x y, y =:|x|V|y|V x, y V .Proof.(i)Vis aR-vector Space : If eitherx= 0 ory= 0 the inequality isobvious. Assume therefore thatx6= 0 andy6= 0. Then we can define x=x|x|Vand y=y|y|Vand obtain| x|V=| y|V= 1. Observing that0 x+ y, x+ y = x, x + 2 x, y + y, y = 2 x, y + 20 x y, x y = x, x 2 x, y + y, y = 2 x, y + 2the claim follows by combining the two inequalities.

5 (ii)Vis aC-vector Space : We can again assume thatx6= 0 andy6= 0 anddefine xand yas above to obtain0 x+ y, x+ y = x, x + x, y + y, x + y, y 0 x y, x y = x, x x, y y, x + y, y which implies|< x, y | 1 since x, y + y, x = 2< x, y . It always is that x, y =r ei for somer 0 and [0,2 )and, from this, it follows that| x, y |=| e i x, y |=|< e i x, y | 1since e i x, y is real and|e i x|= 1. 128 CHAPTER 8. Euclidean Space AND Metric SPACESR emarks (a) IfVis anR-vector Space and , is an inner producton it, we obtain x, y =14(|x+y|2V |x y|2V), x, y Vfor| |Vdefined by|x|V= x, x.]

6 (b) IfVis anC-vector Space and , is an inner product on it, we obtain x, y =14(|x+y|2V |x y|2V+i|x+iy|2 i|x iy|2), x, y V(c) Let(V, | )be an inner product Space . Then the followingparallelogramidentity|x+y|2V+|x y|2V= 2(|x|2V+|y|2V), x, y Vholds.(d) On a normed vactor Space (V,| |V) there exists an inner product , such that|x|V= x, x iffthe parallelogram identity holds (V, , )be an inner product Space . Then|x|V= x, x , x Vdefines a norm need to verify the validity of conditions (n1)-(n3).(i) (n1) follows from (i3).(ii) As for (n2) we have| x|V= x, x = x, x = | |2 x, x =| ||x|V x V K.

7 (iii) Finally the triangular inequality follows from Cauchy-Schwarz. In fact|x+y|2V= x+y, x+y =|x|2V+ 2< x, y +|y|2V |x|2V+ 2|x|V|y|V+|y|2V=(|x|V+|y|V)2 x, y V .Here we used that<z |z|for anyz C. Topology of Metric Open SetsWe now generalize concepts of open and closed further by giving up thelinear structure of vector Space . We shall use the concept of distance inorder to define these concepts maintaining the basic intuition that openshould amount to every point having still some Space TOPOLOGY OF Metric SPACES129 Definition (Open Ball)Let (M, d) be a Metric Space andr (0, ) Then theopen ball aboutx Mwith radiusris defined byB(x, r) :={y M|d(x, y)< r}.

8 Definition (Open Sets)(i)O Mis calledopenor, in shortOo M, iff x O r >0 B(x, r) O .(ii) Any setU Mcontaining a ball B(x, r) aboutxis calledneighborhoodofx. The collection of all neighborhoods of a given pointxis denoted byU(x).Remark collection M:={O M|Ois open}is a (Induced/Relative Metric )Let(M, d)be a Metric Space andN M. Then(N, dN)is a Metric spacewithdN:N N [0, ),(x, y)7 d(x, y).ThenOo NiffO=N Ofor some Oo M, thenOo NiffOo = : IfOo Nandx O, then we findrx>0 such thatBN(x, rx) ={y N|d(x, y)< rx} O .Defining Ox= BM(x, rx) we obtain an open set inM. Setting O= x O Oxo Mwe arrive at O N=O.]

9 = : IfO=N Ofor some Oo M, then for anyx Owe findrx>0such that BM(x, rx) O. In this caseBN(x, rx) = BM(x, rx) N O N=Owhich shows thatOo N. 130 CHAPTER 8. Euclidean Space AND Metric Limits and Closed SetsDefinitions (M, d) be a Metric Space and (xn)n N MN. Thenwe define(i)xn x (n ) >0 N (xn, x ) n N.(ii) A pointxis calledlimit pointof the sequence (xn)n N MNif there isa subsequence (nj)j Nof (n)n Nsuch thatxnj x(j ).(iii) A pointxis calledlimit pointof the the setA (x, r) A6= r > (x, r) :={y M|d(x, y)< r , y6=x}.(iv) A setA Mis closed iff LP(A) A.(v) Theclosure Aof a setA Mis given by A=A LP(A).

10 In this case Ais closed.(vi) A setA B MisdenseinBiffB (M, d)be a Metric Space . ThenA= A Aco M .Proof. = : Letx Aco M. Then we findr >0 such that B(x, r) Ac,which meansx / LP(A). Rephrasing we obtainLP(A) (Ac)c=Awhich givesA= A. = : LetA= Aandx Ac. Thenxcannot be a limit point ofA, thatis, there isr >0 with B(x, r) A= , which amounts to B(x, r) Ac. CompletenessLet (M, d) be a Metric Space andx MN. The sequencexis calledCauchyiff >0 N (xn, xm) m, n N .Any convergent sequence is Metric Space (M, d) is calledcompleteiff every Cauchysequence has a limit TOPOLOGY OF Metric SPACES131 Theorem a sequence(xm)m NinRnbe given.