Chapter 3 The Space of Continuous Functions

1 Chapter 3 The Space of Continuous Functions y r Y W _ { b _ &b "0` fi J [ J " W u~ q uua 4F2 0 9 D |0 | J ) _ 5 yfl r fl { flIn this Chapter we study the Space of Continuous Functions as a prototype of infinitedimensional normed spaces. In Section 1 we review these spaces. In Section 2 the notionof separability is introduced. A proof of Weierstrass approximation theorem differentfrom the one given in Chapter 1 is present in Section 3, following by the general Stone-Weierstrass theorem. The latter is applied to establish the separability of the Space ofcontinuous Functions when the underlying Space is compact.]}}

2 Ascoli-Arezela theorem,which characterizes compact sets in the Space of Continuous Functions , is established inSection 4. Finally in Section 5 we study complete metric spaces. Baire category theorem isproved and, as an application, it is shown that Continuous , nowhere differentiable functionsform a set of second category in the Space of Continuous Spaces of Continuous FunctionsWe studied Continuous Functions on an interval in MATH2050/60 and in a domain boundedby curves/surfaces inR2orR3in MATH2010/20. In this Chapter we will focus on thespace of Continuous Functions defined on a metric (X) denote the vector Space of all Continuous Functions defined onXwhere (X,d)is a metric Space .

3 Recall that in the exercise we showed that there are many continuousfunctions inX. In general, in a metric Space such as the real line, a Continuous functionmay not be bounded. In order to turn Continuous Functions into a normed Space , we need12 Chapter 3. THE Space OF Continuous FUNCTIONSto restrict to bounded Functions . For this purpose letCb(X) ={f:f C(X),|f(x)| M, x Xfor someM}.It is readily checked thatCb(X) is a normed Space under the sup-norm. From now on,Cb(X) is always regarded as a metric Space under the metric induced by the order words,d (f,g) = f g , f,g Cb(X).Some basic properties ofCb(X) are listed below:Property (X)is a complete metric , let{fn}be a Cauchy sequenceinCb(X).

4 For >0, there exists somen0such that fn fm < /4 for alln n0. In particular, it means for eachx,{fn(x)}is a Cauchy sequence inR. By thecompleteness ofR, the limit limn fn(x) exists and we definef(x) limn fn(x).Assuming thatf Cb(X), by takingm in the inequality above, we immediatelyobtain fn f /4< ,hencefn finCb(X). To show thatf Cb(X), weletm in|fn(x) fm(x)|< /4 to get|fn(x) f(x)| /4 for allxandn get|f(x)| |f(x) fn0(x)|+|fn0(x)| /4 + fn0 ,hencefisbounded. On the other hand, asfn0is Continuous , for eachxwe can find a such that|fn0(y) fn0(x)|< /4 wheneverd(y,x)< .It follows that for ally, d(y,x)< ,|f(y) f(x)| |f(y) fn0(y)|+|fn0(y) fn0(x)|+|fn0(x) f(x)| 3 4<.

5 From this proof we see that the completeness ofCb(X) is inherited from the completenessofR, so the underlying spaceXdoes not play any role in this (X) =C(X)whenXis a compact metric need to show everycontinuous function on a compact set is bounded. Assume on the contrary that for somecontinuousf, there are points{xk}such that|f(xk)| . By compactness, there isa subsequence{xkj}andz Xsuch that limj xkj= , by continuity we wouldhave limj |f(xkj)|=|f(z)|< ,contradiction (X)forms an algebra under pointwise that an algebrais a vector spaceVin which a product satisfying the association law is well-definedbetween two points.

6 The interaction between this product and the vector Space structureis reflected in the rulesw(u+v) = (u+v)w=wu+wvand (au)(bv) =ab(uv) for allu,v,w Vanda,b R. It is clear the product of two bounded, Continuous Functions isagain a bounded, Continuous function, soCb(X) forms an will investigate various properties of the spaces of Continuous Functions . Recall thata consequence of Weierstrass approximation theorem tells that every Continuous functionon [a,b] can be approximated by polynomials with rational coefficients. In general a setEin a metric Space is called adense setif its closure is equal to the Space . Thus thecollection of all polynomials with rational coefficients forms a dense set inC[a,b].

7 Sincethis set is countable, we know that every Continuous function in [a,b] can be STONE-WEIERSTRASS THEOREM3from Continuous Functions chosen from a countable set. Our question is, in a general spaceC(X), when does this property still hold? To obtain a result in the positive direction, weneed to establish a generalization of Weierstrass approximation theorem, namely, Stone-Weierstrass theorem. That is what we are going to do in the next Stone-Weierstrass TheoremIn Chapter 1 we proved Weierstrass approximation theorem. Here we first present analternate and optional proof of this theorem. Our proof in Chapter is via a short the proof is a bit longer but (Weierstrass Approximation Theorem).

8 Letf C([a,b]). For every >0, there exists a polynomialpsuch that f p < . the interval to be [0,1] first. Letfbe a Continuous function on [0,1]. Bysubtracting it from a linear function (a polynomial of degree 0 or 1) which passes (0,f(0))and (1,f(1)) we may assumef(0) =f(1) = it to be a Continuous function inRwhich equals zero outside the unit interval and denote the extended function still we approximatefby introducingpn(x) := 1 1f(x+t)Qn(t)dt, x [0,1],whereQn(x) :=cn(1 x2)n.(In fact, asfvanishes outside [0,1], the integration is in factfrom xto 1 x.)Qnis a polynomial and the normalizing constantcnis chosen so that 1 1Qn= 1, in other words,cnis given byc 1n= 1 1(1 x2) will need an upper estimate forcnin a second, 1 1(1 x2)ndx= 2 10(1 x2)ndx 2 1/ n0(1 x2)ndx 2 1/ n0(1 nx2)dx=43 n>1 n,4 Chapter 3.

9 THE Space OF Continuous Functions after using Bernoulli s inequality in the form (1 x2)n 1 nx2for allx. It follows thatcn< this estimate we obtainQn(x) n(1 2)n, x [ ,1],so, in particular,Qn(x) 0 uniformly on [ ,1] for any fixed (0,1).By a change of variables, it is clear thatpn(x) = 1 x xf(x+t)Qn(t)dt= 10f(t)Qn(t x)dtis a polynomial. We claim that pn f , given any >0, there exists such that|f(x) f(y)|< ,|x y|< , x,y [0,1].We have, for allx [0,1],|pn(x) f(x)|= 1 1(f(x+t) f(x))Qn(t)dt 1 1|f(x+t) f(x)|Qn(t)dt 2M 1Qn(t)dt+ 2 Qn(t)dt+ 2M 1 Qn(t)dt 4M n(1 2)n+ 2< ,whereM:= sup|f|,for all sufficiently a Continuous functionfdefined on [a,b], the functiong(x) =f((b a)x+a)belongs toC[0,1].

10 For >0, there exists a polynomialpsuch that g p < .Notingthat the functionq(x) =p((x a)/(b a)) is again a polynomial, we have f q = g p < .So far we have shown that trigonometric Functions and polynomials are dense in thespace of periodic, Continuous Functions and the Space of Continuous Functions respec-tively. In this section we will establish a far-reaching generalization of these results in thespace of Continuous Functions defined in a compact metric Space . In such a Space STONE-WEIERSTRASS THEOREM5trigonometric Functions and polynomials are not available, so we need to seek a reason-able formulation.