Question

12. Let M be the set of continuous functions on R which vanish outside a finite interval (the interval may depend on the function). (a) Show that M is a metric space in the sup norm. (b) Show that M is not complete.

Answer 1

2R R let fem gem let • JCf)= fxe RR : fcos #0} bet Define llflla a sup I fros) 6 het dlf, g) = 11f-olla allere fod : M x (1) Since 1f-gloo de fig) 70 If fr g=o fog ► defra) = 0 If de fig) so 711f-gilio ner NERI HAM => fa 90 20 #netre » c - 2(x) = heº .. (in d (6,5) = sup / (frem - 9n)| he hy gew) - Soml 2 d. (gif) (il let hem news, fent-ho tu frasa jenitt ganto Geast. net 1 fem - hlm 1 = ffin- g(n) + g(n) = helm/ frer Ilona 9147) + gen)-Gemol >> Ifra-8 cm s sebord I few – g(a)! / 969) -hal ynek & sup I few home I & dlfig) + d( gia) a) dofta) e de fig) + d (0,4). I n .:(id) mehre space. TS

лә dева. – lu ft. и душм / ош д», for (n) a feh if nf [0, n] L o elsearten Now, de fn . fm) = sup lfon (or fm (m) / - NER" 1 ) 1 :1Ice if noms N Where for ezo , 7 NEIN .. {fns is caueby sequence in but fem = lim fn (m) = 2 7 if he too) Lo else wler. Sime fin) to for neloia) and [oro) is not finite internal, so fem) & M . 3. M is not complete. e I will upload the next quee part of this fear minutes.

ces Def: A fanefron fir-> IK (obese lk is either p o is said to be Vanishing at infinity if for any &yo. I a compact set k in TR such that Ifm149, tak R2 = continuous the set of all bounded the valued funefron on R . Les fe Cerg. then for at I a compact K nr such that I fall as, to & & xfk : lef Me!= sup flasy Tab Then I flm) 2 max 4, M, } N R R IR >f is old and hence fe CCR) Then Colie) < (R) and we know that CoR) is a subspace of complete mefore space CCR). We show that COCOR) is a closed subspace of CCR). Det & fug be a sequence in Co CRR) convergrey to f in CCR). Then for any &xo , 7 PEIN such that Il fn- flla = sup 1 fn (m) - flas/ ee & nye => 1 fa(m) - femal sex ^A, tuftR In particular, fp (ma femol ce + n), p, knee Now, foln & CoR) » 7 a compact set ik ink Sulh that I foluit ce trek. ::1 FCx) fpras - fla) + 1 focal . <&* & * * * 9 fim e CCR). ! CoCPP) is cosed in COR) : HER

CCRR) is closed subspace of complete mehre spau e CP) Нели Со Се) к со мрlаl. , 1 (d) to show that he is dence in CCR) for this let fe Co (IR) and ayo be any number : Then I a compact set k in IR such that I from Ice & nak Define gir a continuous function gilr [..] Such that g(x) = 1 #kek and g(n)=0,* €& I (9) Are where I(g) = {RER : fcm) # of ist since IR is locally a clearly ke rcg) I compact and Ty space the given_proklengsiu I09) is froste watervol, geri Let honla froggen) Stuen ICA) = ICfg) 108) Karl-dewgo eta then hom en : 'hem fengia Now L =0 H&$109) "Ilf-elle sue | froni Pons goos 5 Surf Praylın gewyt! XPE se the le 7 fem - . CCR) EM and also M s CCIR). á ñ e CCR) = Co CR.) [ CR2) A closed] COCIR) M o dense in CCR). on themsil Cut + 1901)} land (1-1)} 1:43(0)51 feng see