The Real Numbers. RATIONAL FIELD Take rationals as given. is a field with addition and multiplication defined. BOUNDS. Addition: xy= yx, xy z=x yz,

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1 MAT337H Itrodutio to Rel Alysis The Rel Numers RATIONAL FIELD Tke rtiols s give { m, m, R, } is field with dditio d multiplitio defied Additio: y= y, y z= yz, There eists Q suh tht = For eh Q there eists y Q suh tht y= ( y= ) Multiplitio: y= y, y z= yz, There eists Q suh tht = = Q For eh Q { } there eists y Q suh tht y= ( y= = ) Distriutio: yz =yz Aioms of Order If S is set, order o S is reltio deoted y < suh tht If, y S the etly oe of the sttemets y, y, = y is true 2 If y d yz, the z We lso write y if y ; y if y or = y ; y if y or = y Defiitio: Ordered Field A ordered field is field F whih is lso ordered set suh tht (i dditio) y z yz z, 2 If d y the y BOUNDS Defiitio: Bouded Aove Suppose S is ordered set d W is suset We sy W is ouded ove if there eists S suh tht for ll W, Note: eed ot elog to W Defiitio: Lest Upper Boud Suppose S is ordered set d W is suset whih is ouded ove Suppose there eists S suh tht is upper oud for W 2 Wheever is upper oud for W, the The is lled the lest upper oud of W Note: is uique Defiitio: Lest Upper Boud Property A order set S hs the lest upper oud property if wheever W is suset whih is ouded ove, W hs lest upper oud of 35

2 MAT337H Itrodutio to Rel Alysis Note: Q does ot hve this property, ut R does Note: The lest upper oud of W eed ot elog to W If it does, the W hs lrgest elemet or mimum Remrk The order property together with the lest upper oud property hrterize the rel umers Remrk A ordered set with the lest upper oud property lso hs the gretest lower oud property, ie y suset whih is ouded elow hs gretest lower oud Cosider W ={ W } W is ouded ove, so it hs lest upper oud is gretest lower oud for W DEDEKIND CUTS Defiitio: Dedekid Cuts Cosider ll susets S of rtiols suh tht S is ot empty 2 S is ouded ove ut does ot oti its lest upper oud 3 If s S d rs, the r S Note: These sets re i - orrespodee with the rel umers Note: A rtiol umer q is idetified with the set q *={r Q rq } Order If d re Dedekid uts, we defie if is proper suset of Lest Upper Boud Property A set of Dedekid uts is ouded ove if there is ut suh tht for ll The lest upper oud of suh set is just the uio of ll sets i Thus the lest upper oud property is stisfied Field Opertios If d re Dedekid uts, defie ={rs r, s } elemet: *={r Q r} elemet: *={r Q r} The egtive of ut : ={s Q r, r Qsuh tht s r } Deiml Epsio A Dedekid ut e idetified with ifiite deiml epsio Choose iteger suh tht is upper oud for, ut is ot There is uique suh iteger There is uique iteger {,,,9} suh tht is upper oud for, ut Similrly, there is uique iteger 2 {,,,9} suh tht 2 is upper oud for, ut 2 is ot Proeedig idutively, we ostrut ifiite deiml epsio epsio ssoited to Dedekid ut Note: Deiml epsios e tke s the sis for the ostrutio of the rel umers; however, it must e oted tht is ot 2 of 35

3 MAT337H Itrodutio to Rel Alysis suh epsios re ot uique (e: =999 ) Emple I the system of Dedekid uts, 2 is ssoited with the ut {r Q r 2 2or r}, d 2=44 SEQUENCES Defiitio: Sequee A sequee of rel umers is set of rel umers ideed y the turl umers It is usully writte s { } = Defiitio: Covergee The sequee { } overges to the limit L R if give y there eists N suh tht L wheever Remrk A overget sequee must e ouded, ie there eists B suh tht B for ll Defiitios A sequee { } = of rel umers is iresig if A sequee { } = of rel umers is stritly iresig if A sequee { } = of rel umers is deresig if A sequee { } = of rel umers is stritly deresig if Mootoe mes either iresig or deresig A iresig sequee whih is ouded ove is overget The limit is the lest upper oud of the sequee (osidered s set of poits) Note: If sequee { } = is iresig, the either it is ouded ove (d thus hve limit), or else it teds to (ie for ll M there eists depedig o M suh tht M for ) Nested Itervls Lemm Let I =[, ], =,2,3, e sequee of losed itervls suh tht I I The I is ot empty i= Note: This is ot eessrily true for ope itervls For emple, if I =,, the the itersetio is empty Note: This is ot eessrily true for semi-ifiite itervls For emple, if I =[,, the the itersetio is empty Bolzo-Weierstrss Every ouded sequee { } = of rel umers hs overget susequee { k } k= Cotiued Frtios 3 of 35

4 MAT337H Itrodutio to Rel Alysis Cosider Terms re positive = 3 2 = = The differee hge sigs d lso pproh s The susequee of odd terms {, 3, 5, } dereses d is ouded elow The susequee of eve terms { 2, 4, 6, } ireses d is ouded ove So the gretest lower oud of the odd terms must e equl to the lest upper oud of the eve terms, d thus the full sequee mush overge to this umer Sie the deomitor > 9, 2 Let Sie we kow the sequee hs limit L, we hve L= 3L L2 3 L = L= 3±3 2 Sie L we must hve L= 33 2 Defiitio: Cuhy Sequee A sequee { } = is lled Cuhy if for y there eists N suh tht m for ll m,n A sequee { } = (of rel umers) is Cuhy if d oly if it is overget Note: This theorem does ot work with the rtiol umers, ie Cuhy sequee of rtiol umers eed ot overge to rtiol umer CARDINALITY Defiitio Two sets hve the sme rdility of there is -to- orrespodee etwee the poits of the sets Defiitio A hs rdility less th B if there is -to- mppig from A ito B I this se B hs rdility greter th A Defiitio: Coutle A set S is sid to e outle if it e put ito -to- orrespodee with the set of the turl umers N Topology of R Defiitio R {=,, j R} Ier Produt 4 of 35

5 MAT337H Itrodutio to Rel Alysis The dot produt or ier produt is give y, y = j y j j= Notie tht, y is ilier,, y = y,,, d, = = Defiitio: Norm The orm of R is =, Properties: d = = ; = Shwrz Iequlity, y y Proof: If t R the t y,t y Now t y,t y =, 2t, y t 2 y, y = 2 2t, y t 2 y 2 For fied d y, this is qudrti polyomil i t If At 2 BtC t, the 4 B 2 4 AC So 4, y 4 2 y 2, y y Trigle Iequlity y y Proof: Note tht Now tke squre root y 2 = y,y =, 2, y y, y = 2 2, y y y y 2 = y 2 Defiitio: Diste We defie the diste etwee two poits, y R y dist, y = y CONVERGENCE AND COMPLETENESS OF R N Defiitio: Covergee A sequee { k } k= overges to R kn if wheever is give, there eists N suh tht y wheever Lemm A sequee of poits { k } k= i R overges to R if d oly if lim k, j = j k k = k,, k, 2,, k, d =, 2,, j=,, where Proof: It follows from the ft tht if y= y,, y the y j y m y j j=,, Defiitio: Cuhy A sequee of poits { k } k= j, kn i R is Cuhy if for y there eists N =N suh tht j k wheever 5 of 35

6 MAT337H Itrodutio to Rel Alysis Lemm A sequee of poits { k } k= i R is Cuhy if d oly if eh ompoet is Cuhy, ie { k j } k= for eh j=,, : Completeess for R R is omplete, ie y Cuhy sequee overges Proof: If { k } k= is Cuhy sequee, so is { k j } k= for eh j=,, Sie R is omplete, there eists j R suh tht k, j j s k for eh j=,, Hee k s k CLOSED AND OPEN SETS IN N-DIMENSIONAL SPACE Defiitio: Limit Poit Let A e suset i R A poit R is lled limit poit of suset A of R m if there eists sequee { k } k= poits i A whih overges to I prtiulr y poit A is limit poit of Defiitio: Closed Set A set i R whih otis ll its limit poits is sid to e losed Note: Geerlly, set defied y iequlities whih re ot strit re losed (eg: { R 2} ) Propositio The uio of fiite umer of losed sets is losed Proof: Suppose { k } k= A A 2 A m where A,, A m re losed Suppose k R At lest oe of the A,, A m, sy A j must oti ifiitely my terms of the sequee Arrge these terms i order so tht the susripts re iresig, the we get susequee { k l } l= of the origil sequee This susequee overges to Sie A j is losed, A j d hee A A 2 A m So the uio is losed Defiitio: Closure Let A e ritrry suset of R The losure of A, deoted A, is the set osistig of ll limit poits Note: A A Propositio The losure of set A is losed set Proof: Let { k } k = e sequee otied i A d suppose k R Sie k A, there eists sequee of poits i A overgig to k Choose elemet of this sequee, lled k, suh tht k k The lim k k =lim [ k k k ]=lim k = Hee k k A Defiitio: Ope Bll The ope ll of etre d rdius r is the set B r ={ R r } Defiitio: Ope Set 6 of 35

7 MAT337H Itrodutio to Rel Alysis A suset A of R is ope if wheever A there eists r=r suh tht B r A Note: Geerlly, set defied y strit iequlities re ope (eg: B r ) A set A R is ope if d oly if A is losed Proof: Suppose A is ope Let A The there eists r suh tht B r A Let { k } k = e sequee i A The k r So { k } k= ot overge to So is ot limit poit of A 2 Suppose A is ot ope The o ll etred t is otied i A Hee for ll N, we fid poit k B /k suh tht k A where k Hee A is ot losed Propositio The itersetio of fiite umer of ope sets is ope Proof: If U U k d eh U j is ope, the for eh j=, 2,, k there eists r j suh tht B r j U k Let r= mi r j j k The Br U j j d so B r j= k U j Propositio If {U } J is y fmily of ope susets of R, the J U is ope Proof: If U J, the U for some So there eists r suh tht B r j U The B r j U J COMPACTNESS Defiitio: Comptess A suset A of R is ompt if every sequee { k } k= A of poits i A hs susequee whih overges to poit i Propositio The Bolzo-Weierstrss remis true i R Proof: For the se =2 Let k = k, y k e ouded sequee i R 2 The { k } k= sequees of rel umers So { k } k = hs overget susequee Cosider {y k j } j= implies tht this hs further susequee whih is overget, sy {y k jm} m= euse it is susequee of overget sequee Thus k jm = k jm d { y k } k= re ouded The Bolzo-Weierstrss Now { k jm} m= is lso overget, y k jm is overget susequee of poits i R 2 Heie-Borel A suset of R is ompt if d oly if it is losed d ouded Proof: ( ) If the set C is uouded, the for eh k N there eists k C suh tht k k ; suh sequee ot hve overget susequee If C is ot losed, the there eists sequee { k } k= of poits i C whih overges to 7 of 35

8 MAT337H Itrodutio to Rel Alysis poit C ; suh sequee ot hve susequee whih overges to poit of C ( ) Let { k } k= e sequee of poits i the set A Blzo-Weierstrss implies there is overget susequee, d A losed implies the limit of the susequee is i A Ctor's Itersetio If C C 2 C 3 is deresig sequee of o-empty ompt susets of R, the k C k Proof: For eh k there eists poit k C k We hve k C k sie C C k The poits { k } k= forms sequee whih lies i C C is ompt, so there is susequee { k j } j= whih overges to poit C We will show C k k d hee k C k Cosider C m It is possile tht,, m do ot lie i C m, ut ll remiig terms i the susequee { k j } j= do lie i C m The remiig terms form sequee of poits i C m whih overges to By omptess susequee of this sequee overges to C m We must hve = d C m m Hee k C k Ctor Set The Ctor Set is ostruted s follows: S =[,] S = [, 3] [ 2 3, ] S 2 = [, 9] [ 2 9, 3] [ 2 3, 7 9] [ 8 9, ] Cotiue with =3, 4, 5, S is ompt for eh Sie S S 2 S 3, S y Ctor's Itersetio = I some sese, the Ctor Set is smll S otis 2 itervls of legth 3 the middle third of eh itervl gets removed d hs legth 3 2 The sum of the legths of the itervls removed is /3 = = 3 2/3 = I other sese, the Ctor Set is lrge ; it is uoutle d hve the sme rdility s the rel umers Cosider the terry epsio of umer i [,], t k t k= 3 k k =,, 2 (eg 22222) A umer is i S if the first digit is either or 2 A umer is i S 2 if the seod digit is either or 2 So umer i [,] elogs to Ctor's Set preisely whe the terry epsio otis oly 's d 2's Now tke the iry epsio of y umer i [,] (eg ) d mp this poit to the poit with terry epsio y replig eh i the iry epsio y 2 This gives - mp of the itervl [,] ito the Ctor Set Futios LIMITS AND CONTINUITY Defiitio: Limit Let S e suset of R Suppose F : S R m is futio Suppose is luster poit (ie limit poit of S { } We 8 of 35

9 MAT337H Itrodutio to Rel Alysis sy lim F =v where v R m d S if give y there eists suh tht F v wheever Remrk If F = f,, f m where f j : S R, j=,, m, the lim F =v lim f j =v j where v=v,,v m Defiitio: Cotiuity Let S R d let F : S R m We sy tht F is otiuous t if for ll there eists suh tht F F wheever d S Note: If is isolted poit (ot luster poit) of S, the y futio o S is otiuous t Otherwise, F is otiuous t if d oly if lim F =F Defiitio: Lipshitz A futio F i suset S R is sid to e Lipshitz if there is ostt C suh tht F F y C y for ll, y S Note: This oditio implies otiuity Give, the oditio for otiuity is stisfied with = C Propositio A lier mppig from R to R m is Lipshitz I olum ottio, lier mppig is A j j y j j= Proof: A A y=a y = Note tht j= m j j y j j= 2 A A y 2 = i j j y j i j 2 i= j= i= m j= 2 i j j y j j= i j 2 j= = j= j m j j= j j j y j So 2 y 2, d hee A A y C y where C 2 = m j= i= i j 2 Propositio I oe vrile, differetile futio with ouded derivtive is Lipshitz Proof: f f y = f ' t y where t is some poit etwee d y (Me Vlue ) So f f y = f ' t y C y where C is oud for f ' t o the itervl we re osiderig Remrk A futio with otiuous derivtive is lolly Lipshitz Remrk Lolly Lipshitz implies otiuity 9 of 35

10 MAT337H Itrodutio to Rel Alysis Equivlet Coditios For Cotiuity Let F : S R R m F is otiuous t S if d oly if for ll sequees { k } k = hve F k F s k of poits i S suh tht k we Proof: Let S d let { k } k= e sequee i S whih overges to Let Sie F is otiuous t, there eists suh tht F F whe for S Sie { k } k = overges to, there eists N suh tht k whe kn So F k F whe kn Hee F k F For the overse, it is equivlet to show tht if F is ot otiuous the the sequee oditio is ot stisfied If F is ot otiuous, the for ll there eists suh tht there is poit = suh tht ut F F Costrut sequee { k } k= i S overgig to s follows Tke = k There eists k S suh tht k k ut F k F The k s k ut F k does ot overge to F Defiitio: Reltively Ope Let S e suset of R A suset V of S is (reltively) ope if there is ope suset U of R suh tht U S =V Let F : S R R m F is otiuous if d oly if wheever W is ope suset of R m, F W { S F =w W } is ope suset of S Proof: Suppose F is otiuous Suppose W is ope suset of R m Let F W, ie F =u W Sie W is ope, so there eists suh tht B u W Sie F is otiuous, there eists suh tht for ll S suh tht, we hve F F = F u This implies F W Hee F W otis B S, d therefore F W is ope For the overse, let S Let Cosider B F =W ope set i R m Sie F W is ope suset of S otiig, so there eists suh tht B S F W This sys if S d the F F PROPERTIES OF CONTINUOUS FUNCTIONS Properties of Limits of Cotiuous Futios Let f : S R R m d g : S R R m e defied o S R Let S Let lim Let R The: lim lim f g =uv f =u If f d g re rel vlued, the lim If v, the lim f g =u v f g =u v f =u R m d lim g =v R m Note: Coorditig futios i R re otiuous; tht is if =,, the j is otiuous futio of Therefore y polyomil i,, is otiuous futio of Ad hee y rtiol futio i,, is otiuous wheever the deomitor is ot zero of 35

11 MAT337H Itrodutio to Rel Alysis Compositio Property The ompositio of otiuous futios is otiuous Suppose f : S R R m Let T e suset of R m otiig the rge of f Suppose g :T R m R k The g f = f g is otiuous if f d g re Tht is if f is otiuous t d g is otiuous t f, the g f is otiuous t Proof: Let { j } j= e sequee of poits i S overgig to S The { f j } j= overgig to f sie f is otiuous The {g f j } j= ={g f j } j= g f =g f sie g is otiuous is sequee of poits i T is sequee of poits whih overges to COMPACTNESS AND EXTREME VALUES If K is ompt suset of R d f : K R m is otiuous futio, the f K is ompt Proof: Let { y k } k= K There eists susequee { k j } j= f, therefore f K is ompt e sequee i f K Choose k K suh tht f k = y k for k=, 2, { k } k= whih overges to poit K Sie { f k j } j= ={ y k j } j= is sequee i overges to Etreme Vlue If K is ompt suset of R d f : K R is rel vlued otiuous futio o K, the f ssumes mimum d miimum i f K, ie there eists poits, K suh tht f f f K Proof: K is ompt, so f K is ompt, or equivletly, losed d ouded Thus f K hs lest upper oud (supremum) M d gretest lower oud (ifimum) m, ie m f M There eists sequee of poits i f K overgig to M, d sie f K is losed M f K Hee there eists K suh tht f =M Similr for m UNIFORM CONTINUITY Defiitio: Uiform Cotiuous Let S R A futio f : S R m is uiformly otiuous if for ll, there eists suh tht for ll S f f wheever d S Remrks A futio whih is Lipshitz, ie f f y C y, y S, is uiformly otiuous 2 A lier trsformtio from R to R m is Lipshitz, hee uiformly otiuous A otiuous futio o ompt set is uiformly otiuous Proof: Suppose this is ot true The there eists K R ompt, f : K R m whih is ot uiformly otiuous So there eists suh tht there is o suh tht for ll K, f f wheever d K With this, tke = k There eists k, k K suh tht k k k ut f k f k Sie { k } k= of poits i K (ompt), there is susequee { k j } j= whih overges to poit K Sie is sequee of 35

12 MAT337H Itrodutio to Rel Alysis k j k j k j, j=, 2,3,, so k j lso s j f is otiuous t, hee f k j f But f k j f k j Cotrditio Hee f must e uiformly otiuous THE INTERMEDIATE VALUE THEOREM Itermedite Vlue If f is otiuous rel-vlued futio o [,], the f ssumes every vlue etwee f d f o [, ] Proof: It is suffiiet to show tht If f is otiuous o [,] d if f d f, the there eists poit [, ] suh tht f = Let S ={ [, ] f } S sie S S is ouded ove y, hee S hs lest upper oud We'll show f = Suppose f The y otiuity, there eists suh tht f f f if d [,] 2 So f f 2 f f f, i prtiulr, this is true for, provided [, ] Thus is 2 ot upper oud for S Suppose f The y otiuity, there eists suh tht f f f if d [, ] 2 This implies f for, provided [,] Thus is ot the lest upper oud for S The oly possiility left is f = MONOTONE FUNCTIONS Defiitio: Mootoe f :, R is mootoi if it is iresig o, or deresig o, Remrk If f is iresig, the f is deresig, d vie vers Propositio If, d f iresig, the lim f =L d lim f =M -, oth eist d L f M + Proof: Let L= sup f d M = if f,, Let The L is ot upper oud for { f : }, ie there eists, suh tht f L Sie f is iresig, f L for, Let = The f L whe, Thus lim Sie L= sup f,, f f f =L -,, hee sup, f =L f Remrk f is otiuous t if d oly if L= f =M Suppose f is mootoe o losed itervl [, ] The the umer of disotiuities of f is t most outle 2 of 35

13 MAT337H Itrodutio to Rel Alysis Proof: Without lost of geerlity, we ssume f is iresig For y poit [, ], defie the jump of f t to e j =lim f lim f + - ( j = iff f otiuous t ) How my poits there e t whih the jump is greter th? N f f How my poits re there where the jump stisfies 2 j k, k=, 2,3,? Let tht umer e N 2 k k The N k 2 f f, so k N k 2 k f f Every disotiuity t hs the property tht there eists k N suh tht Hee the umer of disotiuity is outle 2 j k 2 k Normed Vetor Spes Defiitio: Normed Vetor Spe A vetor spe V is sid to e ormed if there is futio o V suh tht v, d v = v= 2 For R, v = v 3 uv u v Emple Eve i R, there re other wys to defie orm esides the stdrd Eulide orm The p-orm is defied y p = p p p The se p=2 gives the Eulide orm = =m,,, p where =,, TOPOLOGY IN NORMED SPACES I ormed vetor spe, the orm e used to defie overgee of sequee, Cuhy sequee, ompleteess, ope sets, losed sets, otiuity of futios, et Defiitio: Covergee Let V e ormed vetor spe A sequee { m } m N is sid to overge to poit V if for ll there eists N suh tht m wheever mn Defiitio: Cuhy Sequee A sequee { m } m N is sid to e Cuhy if for ll there eists N suh tht m wheever m,n Defiitio: Completeess V is sid to e omplete if y Cuhy sequee hs limit i V Defiitio: Ope Bll The ope ll with etre V d rdius r is defied y B r ={v V v r } Defiitio: Ope Set 3 of 35

14 MAT337H Itrodutio to Rel Alysis A set A is ope if for ll A there eists r suh tht B r A Defiitio: Limit Poit V i limit poit of set A V if there is sequee i A whih overges to Note: eed ot e i A Defiitio: Closed Set A set is losed if it otis ll its limit poits Emple Cosider sequees of R :, 2, suh tht j= j p This gives ormed vetor spe with orm j= p j p Emple Certi spes of futios give ormed vetor spes Let K e ompt suset of ormed vetor spe Cosider the spe V for otiuous futios o K with the orm f =m f ( otiuous futio o ompt set hs mimum) This orm stisfies the three properties of K orms Remrk The ope uit ll with respet to y orm is ope with respet to y orm Remrk The uit ll B with respet to geerl orm o ormed vetor spe is ove, ie the lie segmet t t y=t y, t is otied i B wheever, y B Proof: If d y, the t t y t t y t t y tt= Remrk Cosider the losed Eulide uit ll B E If, y B E, the the lie segmet joiig d y is i the ope Eulide ll eept possily for the ed poits The losed Eulide ll is stritly ove Note tht the losed uit lls i the mimum orm d orm re ot stritly ove; there re lie segmets i the oudry Emple: Norms o Vetor Spes of Futios Let V e vetor spe of otiuous rel vlued futios o ompt set K We defie the orm to e f =m f K 2 Let V e vetor spe of otiuous rel vlued futios o itervl [, ] Cosider f p =[ f d] p p where p This stisfies the properties of orms 3 Let C k [, ]={ f : [, ] R f d its derivtives up to order k re otiuous o [,]} This is vetor spe We defie f = f f ' f k INNER PRODUCT SPACES 4 of 35

15 MAT337H Itrodutio to Rel Alysis Defiitio: Ier Produt Spe A vetor spe V is ier produt spe if there is rel-vlued futio o V V suh tht Positive defiiteess:, with equlity if d oly if = 2 Symmetry:, y = y, 3 Bilierity: 2 2, y =, y 2 2, y (similrly, y 2 y 2 =, y 2, y 2 ) Remrk I ier produt spe, =, 2 defies orm Emple Let C [, ] e the vetor spe of otiuous rel-vlued futios o [, ] Defie f, g = C [, ] eomes ier produt spe f g d The Cuhy-Shwrtz Iequlity, y, with equlity if d oly if d y re ollier By ovetio, is ollier with y vetor Proof: Let t R The t y, t y, 2t, y t 2 y, y This is qudrti i t, d thus its disrimite is 2, y y 2 Emple The Cuhy-Shwrtz Iequlity is true for ifiite (squre-summle) sequees Let { j } j N d { y j } j N e sequees suh tht d j= j y j j= 2 j 2 j= y 2 2 j j= j 2 d j= Sie series whih is solutely overget is overget d j= j y j j= 2 j 2 j= y 2 j Tkig the limit s N yields j= y j 2 The j= j 2 By the Cuhy Shwrtz Iequlity, j= j y j j= 2 j 2 j= y 2 2 j Hee, the sum of two squre-summle sequees is squre summle; t j 2 =t 2 j= j= produt of this spe N j= j= j y j d j y j re overget j= j, it is suffiiet to show N j y j j= j= 2 j j y j 2 = j= N 2 j= y 2 j 2 j= 2 j 2 j 2 j y j j 2 So the set of squre summle sequees form vetor spe, d, y = j= j= j= 2 j= y 2 2 j y j 2 Also j y j is ier ORTHONORMAL SETS Defiitio: Orthogol Two vetors re orthogol if, y = Defiitio: Orthoorml A set of vetors is orthoorml if they re pirwise orthogol, d v = for eh v i the set 5 of 35

16 MAT337H Itrodutio to Rel Alysis Grm-Shmidt Proess If { j } is fiite or ifiite set of vetors i ier-produt spe, there is wy to ostrut set of orthoorml vetors with the sme sp If =, throw it out If, let y = Let y 2 = 2 2,, = 2 2, y y, y y The y, y 2 = Let y 3 = 3 3, y y, y y 3, y 2 y 2, y 2 y 2 Cotiue idutively Let v = y y, v 2= y 2 y 2, et The { v j} is orthoorml sis with the sme sp s { j } Remrk If e,, e is orthoorml sis of ier produt spe, d if v= e e, the j = v, e j Trigoometri Polyomils d Fourier Series Cosider futios o [,] Defie f, g = f g d The: 2 = 2 2 d = 2 {,2si,2 os } N is (ifiite) orthoorml set Epsios With Respet to Orthoorml Bses Let e,, e e orthoorml set i ier produt spe V Let M e the suspe sped y e,, e The y M e writte (uiquely) s = j e j, where j =, e j Also, if y= j= j= j e j M, the, y = j= j j I prtiulr, 2 = 2 j Note: These fts re ompletely logous to wht hppes whe V =R with the dot produt d { e j } j= is the stdrd sis j= ORTHOGONAL EXPANSIONS IN INNER PRODUCT SPACES Defiitio: Projetio A projetio o vetor spe V is lier mp P :V V suh tht P 2 =P Note: P R P =I where R P is the rge of P, euse P P =P 2 =P Note: If P is projetio the I P is lso projetio, euse I P 2 = I P I P =I P PP 2 =I P Defiitio: Orthogol Projetio P is sid to e orthogol projetio if ker P is orthogol to R P Note: We eed to e i ier produt spe to tlk out orthogol projetio Emple 6 of 35

17 MAT337H Itrodutio to Rel Alysis I R 2,, y, is projetio The -is is the rge, the y -is is the kerels, so this is orthogol projetio I P, y =, y is projetio Projetio Let e,, e e orthoorml set i vetor spe V Let M e the suspe sped y e,, e Defie P :V M y P y= y, e j e j j=, y V The: P is the orthogol projetio oto M 2 y, e j 2 y 2 j= 3 For ll v M, y v 2 = y P y 2 P y v 2 Hee P y is the loset vetor i M to y (this requires P to e orthogol projetio) Proof: P 2 =P is ler P is orthogol projetio Let P = j e j R P=M Let y ker P So j= P y= y, e j e j = y, e j = y, j= j= j e j = y, = Hee ker P R P=M Let P y= j e j, the P y 2 = 2 j Now let = j e j M d y V The j= j= j= y 2 = y, y =, 2, y y, y = j= = j= = j= 2 j 2 j j y 2 j= j 2 2 j= j j 2 j j= j j 2 P y 2 y 2 = P y 2 P y 2 y 2 j= j 2 y 2 If we set =P y, we get P y y 2 = P y 2 y 2 This shows y 2 P y 2 = y, e j 2 Also, we hve y 2 = P y 2 P y 2 y 2 = P y 2 P y y 2 j= Bessel's Iequlity Let S N d let {e S } e orthoorml set i ier produt spe V For V,, e 2 2 S Proof: If S is fiite, this follows from Projetio If S is ifiite, the we might s well tke S=N Cosider N =, e 2 2 (follows from Projetio ) Lettig N gives the result Defiitio: Hilert Spe A Hilert spe is omplete ier produt spe Note: Ay fiite dimesiol ier produt spe over R is omplete 7 of 35

18 MAT337H Itrodutio to Rel Alysis Defiitio: Sp The sp of set of vetors T i Hilert spe, deoted sp T, is the set of fiite lier omitios of vetors i T Defiitio: Closed Sp The losed sp of set of vetors T, deoted sp E, is the losure of sp T Note: This is gi suspe Prsevl's Let E e fiite or outly ifiite orthoorml set i Hilert spe H If E is fiite, sy E={e,, e }, the sp E=sp E If E is ifiite, the the suspe M =sp E osists of ll vetors elogs to l 2 (ie 2 ) = I either se, if H, the sp E if d oly if, e 2 = 2 Proof:, e 2 = 2 Otherwise, E={ e } = suh tht =K follows from Projetio if we hve fiite orthoorml set Let { } = l 2 Let k = k = 2 2 Hee if l k K, l k 2 = e { k } k = l =k 2 e = e where the oeffiiet sequee { } = (ie equlity ours i Bessel's Iequlity) is Cuhy sequee Let There eists K l = =k overges to H Sie M =sp E, M Now suppose H is ritrry Set =, e Bessel's Iequlity implies { } l 2 y= e The y 2 = 2 2, y y 2 = 2 2, e y 2 = 2 2, e 2 = = 2 2 Now y 2 = 2 = 2 =y = e M 2 2, so l k Therefore { k } k= sie 2 2 Let Defie l 2 ={ =, 2, = omplete, hee it is Hilert spe 2 } with ier produt, y = y d orm =, The spe l 2 = Proof: Let { k } k= e Cuhy sequee, with k = k, = So for ll there eists K suh tht k l wheever k,lk Note tht k, l, k l, hee the -th ompoet of the k 's form Cuhy sequee of rel umers Sie R is omplete, there eists y R suh tht k, y s k Let y= y = = y, y 2, Show: y l 2 Note tht k l k l for k,lk Hee the sequee { k } k= N overges to limit L Fi N The y 2 =lim = k Show: k y i l 2 N = y k, 2 =lim l N is Cuhy d so it k 2 =L Let N d get y 2 L Thus y l 2 = k, 2 lim = k orm Fi d hoose K suh tht k l wheever k,lk Fi N The N = l, k, 2 lim l k 2 2 Let Let N d get y k 2 2 y k Sie is k is 8 of 35

19 MAT337H Itrodutio to Rel Alysis ritrry, this sys k y i l 2 orm s k Hee l 2 is Hilert spe THE L P NORMS Defiitio: L p Norm The L p orm o C [, ] is defied to e f p = f d p p where p Remrk f L p [, ] if f d p p is fiite Lemm If A, B d t, the A t B t t A t B Equlity ours for some t, if d oly if A=B Proof: Let, R Let e =A d e =B The sttemet eomes e t e t =e t t t e t e Assume d wlog Sie d e =e, the result follows Hölder's Iequlity (for itegrls) f d p where p If f L p [, ] (ie f p d is fiite) d if O C [,], osider f p g L q [, ] where p q =, the Proof: It is suffiiet to show f p d g q Let A= p g q f p p q g q f p f g f p f g d f p g q f g d f p g q Note tht it is true for f or g, so we ssume f p p g d B=, fied Let t= f p p g p p p, the t= q sie p = The q q f p p f p g q p q g q Tkig of oth sides we get q d= f g d g q f p g f p q p f p d g q p q g q d= f p p q p f p p g q q q g q = q p = Thus q f g d f p g q Geerliztios of Hölder's Iequlity Oe itrodue weight futio ( w positive, otiuous or pieewise otiuous) i the itegrls, ie f g w d w f d p w g d q q A emple is w = 2 used o the itervl p [,], whih leds to system of orthogol polyomils lled the Legedre polyomils 2 The itervl ould e ifiite or semi-ifiite I these ses, oe eeds to osider overgee of the itegrls 9 of 35

20 MAT337H Itrodutio to Rel Alysis 3 f d g ould e pieewise otiuous (otiuous eept for fiite umer of disotiuities if the itervl is fiite, or fiite umer of disotiuities o y fiite suitervl if the itervl is fiite) 4 There is more powerful versio of itegrtio theory, lled Mesure Theory or Leesgue Itegrtio, i whih Hölder's iequlity still holds Hölder's Iequlity (for sequees) If = l p (ie sequees suh tht = p ) d = l q where p =, the q = = p p = q q Note: These iequlities remi vlid with weights w, w 2,, w j, ie = w = w = w p p = w q q Proof: Tke f =, [, d g =, [, for f d g re pieewise otiuous o [, The f p d= p = d g q d= q = Mikowski's Iequlity (for itegrls) (This is the trigle iequlity for L p spes) Let f, g C [, ] d p The f g p d p f d p p Note: This shows the p -orms re ideed orms, ie they stisfy the trigle iequlity Note: This is lso true for p= d p= Proof: Let q =, the q= p p Note tht f p q = Now, f g p p = p f d p q q = = f g p d f d p q f g p f g d = f d p p p f g p f d f g p g d f g d p q f g d p q q p q f p d f p g p p g d p p p q = f p q = f p p f g d p q = f g p q f p g p p Sie q = p, we hve f g p p f g p p f p g p f g p f p g p q g d p p Mikowski's Iequlity (for sequees) 2 of 35

21 Let p The = p p = Note: This is lso true for p= d p= MAT337H Itrodutio to Rel Alysis p p = p p Remrk Similrly, there re versios of Mikowski's Iequlities with weights There re geerliztios to pieewise otiuous futios, ifiite itervls, et Propositio If f C [, ] d, rs, the f r f s Also f p f Note: If the itervl does ot hve legth, the ostt (depedig o r d s ) is itrodued i this iequlity Proof: f r d= f r d Usig Hölder's Iequlity with p= s, we get r f r d f d r p d q q f d r p p, so f r = 2 f p = f d r r f d p p p p r f d r p f p p d = = = f f d s s= f s p d = f Remrks If the itervl does ot hve legth, the ostt (depedig o r d s ) is itrodued i these iequlities 2 O the fiite itervl, overgee i L implies overgee i L p More geerlly, overgee i L s implies overgee i L r o fiite itervl if rs 3 This is o loger true o ifiite itervls O ifiite or semi-ifiite itervl, L r L s d L s L r if rs Limits of Futios Wht is the reltio etwee overgee i L p d overgee i L q for differet vlues of p d q? LIMITS OF FUNCTIONS Defiitio: Poitwise Covergee If { f } = is sequee of rel-vlued futios o [, ], the f f poitwise if for eh [, ], f f s Defiitio: Uiform Covergee If f f i mimum orm, the f is sid to overge to f uiformly Note: f f uiformly if d oly if for ll there eists N suh tht f f for ll [, ] wheever N Remrk 2 of 35

22 MAT337H Itrodutio to Rel Alysis If S is ompt, the uiform overgee is equivlet to overgee i L S Ay otiuous futio o ompt set S is i L S If S is ot ompt, osider the spe C S ={ f : S R whih re otiuous d ouded} For futios i C S, uiform overgee is equivlet to overgee i L S For sequee of futios { f } = C S, f f uiformly if d oly if m f f s S UNIFORM CONVERGENCE AND CONTINUITY Emple If f = o [,], f overges poitwise to f = {, The limit futio is ot otiuous This, = ot hppe if the overgee is uiform If { f } = is sequee of otiuous futios o S d f f uiformly o S, the f is otiuous Proof: Let S d Now f f = f f f f f f, so f f f f f f f f Now hoose suffiietly lrge suh tht f f 3 S (so i prtiulr f f ) With this hoie of, hoose suh tht 3 f f 3 if With this hoie of, f f 3 3 = Hee f is otiuous 3 Completeess The spe C K of otiuous rel-vlued futios o ompt set K R Proof: Suppose { f } = is Cuhy sequee i Fi K The { f } = is omplete is Cuhy sequee of rel umers, so f f m f f m For eh K there eists rel umer f suh tht f f The overgee of f to f is uiform, hee f is otiuous futio Therefore { f f } implies C K is omplete UNIFORM CONVERGENCE AND INTEGRATION Propositio If { f } = is sequee of otiuous futios o [,], the Note: This is ot true i geerl if the overgee is ot uiform Note: This is ot true i geerl if the itervl is ifiite Proof: f d f d = f f whe N For suh N, f d f d f f d f f d Let The there eists N suh tht f f d d= Hee f d f d 22 of 35

23 MAT337H Itrodutio to Rel Alysis Itegrl Covergee Let { f } = e sequee of otiuous futios o [,] suh tht f f uiformly o [, ] Fi poit [, ] Let F f t dt d F f t dt The F F uiformly o [, ] Suppose {h } = is sequee of C futios o [, ] suh tht h ' overges uiformly to futio g o [, ], d suppose there eists poit [, ] suh tht lim h = eists The h overges uiformly to differetile futio h suh tht h = d h'=g Proof: Let h =h h ' t dt Sie h d h h = g t dt This sys h' =g d h = h ' t dt g t dt uiformly, therefore Remrk I omple vrile theory, uiform overgee of sequee of lyti futios i domi does imply uiform overgee of the derivtives o ompt susets Emple si Cosider f = It is ler tht f uiformly for R s However, f ' =os whih does ot overge o R or o y fiite suitervl si I C, does ot overge to Suppose f,t (rel-vlued) is otiuous o [, ] [,d ] Let F = d f,t dt The F is otiuous o [,] Proof: f is uiformly otiuous o [,] [, d ] So give there eists suh tht f,t f 2,t 2 whe,t 2,t 2 I prtiulr f,t f 2,t whe 2 for ll t Now F F 2 = d d F F 2 d f,t dt d f 2,t dt= f,t f 2,t dt, so if 2 the d f,t f 2,t dt dt= d Hee if 2 is fied, F F 2 s 2 Leiiz's Rule: Differetitio Uder Itegrl Suppose tht f,t d f d f o [, ] d F ' =,t dt,t re otiuous o [, ] [,d ] Defie F = d f,t dt The F is otiuous 23 of 35

24 MAT337H Itrodutio to Rel Alysis Proof: Fi [,] d let h R, h Wt F h F h Fi h F d h F = h poit t lyig i etwee d h suh tht f h,t f,t = f d f,t dt s h f h,t f,t dt By the Me Vlue, for eh t [, d ] there eists h t,t h, d so f h,t f,t = f h t,t Note tht MVT does ot sy t depeds otiuously o t, however for fied h the LHS is otiuous futio of t d so the RHS is lso Now f t,t = f,t is otiuous o the ompt set [, ] [, d ], so it is uiformly otiuous So give there eists suh tht f,t f 2,t 2 whe,t 2,t 2 I prtiulr f,t f 2,t whe 2 for ll t Fi h with h d d d d F h F f h,t dt = f h,t f,t f h,t dt f t,t f,t dt dt= d sie t h This implies lim h F h F h d f,t dt=, hee the result Emple d d si y dy= si y dy= os y y os y dy= y si y si y dy= y os y dy = os, so d d si y dy= si y dy= si [ os y 2 ] si os 2 2 = si os 2 2 SERIES OF FUNCTIONS Series of Rel Numers A series of rel umers is = The ssoited sequee of prtil sums is s = The series is sid to overge if the prtil sums hve limit s, d s is sid to e the sum of the series k= k Defiitio: Asolute Covergee A series is solutely overget if overges = A solutely overget series is overget Proof: If is give, the there eists N suh tht wheever N k l (this is the Cuhy riterio of the prtil sums of = equivletly, the prtil sums l ) Hee =k l =k l =k This is the Cuhy riterio for the overgee of =, or 24 of 35

25 MAT337H Itrodutio to Rel Alysis Altertig Series Test If 2 d lim =, d if the sigs of the 's re stritly ltertig, the the series overges = Defiitio: Series of Futios If { f k } k= re futios defied o suset S R Note: We hve the otio of poitwise overgee, uiform overgee, L p with vlues i R (or i R m ), the we osider the series f k= overgee, et Emples Power series: or = 2 Fourier series: f ~A A os B si = = If k = f is series of otiuous futios whih overges uiformly, the the limit is otiuous futio A series of futios is uiformly overget if d oly if it is uiformly Cuhy A sequee of futios is uiformly overget if d oly if it is uiformly Cuhy Proof: If {s } = is uiformly overget, the give there eists N suh tht s s 2 for ll whe N The for m,n d for ll, s s m = s s s m s s s s m s 2 2 = Equivletly, s s m = s s s m s s s s m s 2 2 = Now suppose the sequee is uiformly Cuhy Give there eists N suh tht for ll ive there eists N suh tht for ll S, s s m whe m, N For eh fied, {s } = is Cuhy sequee of rel umers Hee there eists s R suh tht s s s Also s s m whe m,n idepedet of Let m, the s m s Therefore s s whe N Sie N is idepedet of, the overgee is uiform Weierstrss M-Test Suppose { f } = re futios o S R k with vlues i R m, d suppose {M } = re o-egtive rel umers suh tht f =sup f m The if S M, the f overges uiformly o S = = 25 of 35

26 MAT337H Itrodutio to Rel Alysis Proof: Note tht = f k = f = k { f } =k lim k f f lim k = = { f } M, hee f is solutely overget f =k =k M = = f =k = f M =k idepedet of Hee Emple Cosider the power series = 2 2! O the itervl [ A, A] with A, 2 2! A2 overges solutely y the rtio test for y A Hee the power series = 2 2! 2! = A 2 2! overges uiformly o [ A, A] POWER SERIES Root Test Suppose If lim sup = d, the the series overges Note: If re ot o-egtive, we pply the test to = = Proof: Choose ' suh tht ' The ' for suffiietly lrge, d so ' Now ' overget geometri series Hee overges y the Compriso Test = = is Hdmrd's If is power series, the oe of the followig is true = The series overges for = oly 2 The series overges for ll R 3 There eists R suh tht the series overges for R d diverges for R (it my or my ot overge for =±R ) The series overges uiformly for rr Proof: Let =lim sup : lim sup =lim sup,if =, if = We lim the rdius of overgee is R={ = lim sup = If =, the = for ll So the series overges for ll 2 If =, the = uless = So the series overges oly if =,if, Apply the root test to = 3 If =R, the the series overges If =R, the the series diverges If rr the r = overges Sie r =M, the Weierstrss M-Test implies = overges 26 of 35

27 MAT337H Itrodutio to Rel Alysis uiformly o [ r, r ] Suppose = H is series of C eists [, ] suh tht = H = d H '=G futios o [,] suh tht H ' overges uiformly to futio G d there = H = eists The H overges uiformly to differetile futio H suh tht = Applitio to Power Series (Differetitio) If f =, the f ' = = Proof: Let lim sup lim sup =lim sup origil series = Now tke H = = = = with the sme rdius of overgee It is ler tht overges if d oly if = lim lim sup =, hee = = = = overges Now hs the sme rdius of overgee s the o [ r, r ] where rr= The H ' overges to futio G o [ r, r ], d = H = overges to limit (i ft ostt) We olude tht H overges to differetile futio H o [ r, r ] suh tht H = d H '=G ( H = = = d G = = ) Remrk If f = = o R, R, the f k = = k k Also = f! Applitio to Power Series (Itegrtio) If f = =, the F = is the sme s tht of the origil series f t dt= = t dt= = The rdius of overgee of the itegrted series Emple Cosider =, The overgee is uiform o [ r, r ] for y fied r [, (y the Weierstrss = M-test sie r d r overges) There is o overgee t the edpoits F k = k = k t dt= = overges uiformly o [ r,r ] to F = dt=log Notie tht log is ot defied d the series diverges whe = t = t dt= Now = 27 of 35

28 MAT337H Itrodutio to Rel Alysis Whe =, the series = = 2 3 does overge (very slowly) y the ltertig series test Also, 4 log is defied t = ; its vlue is log 2 Is it true tht log 2= 2 3 4? Ael's Let f = = eists d its vlue is o, Suppose lso tht = = overges, ie the series overges whe = The lim f - Approimtio y Polyomils TAYLOR SERIES Suppose f : [, d ]R d f hs derivtives of ll orders Let [, d ] Is f = Two questios: Where does the power series overge? 2 Whe it does overge, does it overge to f or to somethig else? = whe = f!? Note: I omple lysis, if f is lyti i eighorhood of poit C, the f hs Tylor series epsio out whih overges to f i the dis of eter d rdius equl to the diste from to the erest sigulrity of f Emple Cosider f =e We eted f y defiig 2 f ==lim e 2 Oe show tht Tylor series of f out = is 2 d d f = = Hee the Emple: Differetil Equtio Method This is sometimes useful i studyig overgee of Tylor series to the origil futio Cosider where R Rell tht = = g ' = g is vlid i Notie tht g = stisfies, g = Solve this differetil equtio y power series method Try f = oeffiiets re to e determied f ' = f ' = f ' = f = = [ ]= = = = = [ ] So we hve where the = Hee for =,,, = = We kow = sie = whe =, d therefore we kow ll the 's; = 2 =! If is positive iteger (sy =N ), the the series termites d oiides with the epsio of N 28 of 35

29 MAT337H Itrodutio to Rel Alysis Also, lim f = = = =lim = Therefore the rdius of overgee is = solves the differetil equtio f ' = f d stisfies the iitil oditio f f = To show f =, we show tht =ostt The ostt must e sie f = d d = We tke d[ f f ' = ] f = f f = 2 2 Defiitio: Tylor Series If f hve derivtives of ll orders i eighorhood of poit, the Tylor series of f t is k= f k k k! Defiitio: Tylor Polyomil P = k= f k k is the Tylor polyomil of order k! If f C [ A, B ], d if M is oud for f o [ A, B ], d if [ A, B ], the R = f P stisfies R M! Proof: P hs the sme vlues s f for its derivtives up to order t the poit, therefore k R = f k k P for k=,,, Also R = f P = f sie P is polyomil of degree Now y FTC, R =R R t dt= f t dt So R k Suppose we hve show tht for some k, k, R M k k! M dt =M Now R k =R k so R k M t k k! dt k =M By idutio the estimte is true for k=,,, k2! Hee R M! R k t dt, HOW NOT TO APPROXIMATE A FUNCTION The Tylor polyomil use iformtio oly t oe poit Cosider otiuous futio f o [,] Pik,, i [, ] There is uique polyomil of degree suh tht = f j Does this proedure produe fmily of polyomils whih overge to f i the uiform orm? Not eessrily Weierstrss Approimtio Give otiuous futio f o [, ], there eists sequee polyomils { } = suh tht f s Proof: We'll give proof usig Berstei's polyomils Note tht it is suffiiet to prove the theorem whe [,]=[,] 29 of 35

30 MAT337H Itrodutio to Rel Alysis BERNSTEIN'S PROOF OF THE WEIERSTRASS THEOREM Berstei Polyomils Rell y = k= P k = k k k k k y k Set y= The = k k k = Let, k =,,, These re the Berstei polyomils Notie tht: P k is polyomil of degree P k o [,] ; zeros our oly t the edpoits P k hs mimum t k, k=,,, If f is otiuous futio o [,], we defie the polyomil B f = f k= k P k Properties B f is polyomil of degree t most 2 B f g =B f B g 3 B f = B f 4 B f if f 5 B f B g if f g 6 B f B g if f g k= Lemm B = 2 B = 3 B 2 = 2 =2 2 Proof: B = k= P k = 2 Note tht k k = k k B = k= 3 Note tht k 2 B 2 = = k k k = k = k 2 k 2 k= k= k! k! k! =! k! k! = k So k= k k k = k= k k k = =! k! k! =! k 2! k!! k! k! = 2 k=2 k 2 k k k= k k k k k k = 2 2 k k k = k 2 k k= 2 k 2 k So If f C [, ], the B f f uiformly s 3 of 35

31 MAT337H Itrodutio to Rel Alysis Proof: We wt to show for ll, there eists N suh tht f B f for ll N Note tht f is uiformly otiuous o [,], ie give there eists suh tht f f y wheever y d, y [,] Let M = f Fi [,] The f f 2 2 M 2 for ll [,] If, this is true If 2 2, the f f 2 M 2 M 2 2 M Rell tht B u B v if u v So B [ f f ] = B f f B 2 2 M 2 2 Set = The B f f 2 2 M 2 2 = 2 M = 2 2 M The futio = 2 hs its mimum o [,] whe = 2 2 = 4 Therefore B f f 2 2 M 2 with vlue of 4 = 2 M The RHS is idepedet of 2 2 M Choose N suh tht 2 N 2 2 N M 2, the for ll N, 2 M = FOURIER SERIES Defiitio: Fourier Series Let f, g C [,] Let f, g = [,] Defie the Fourier series of f to e f g d, the {,2os, 2si } N is orthoorml system o f ~ A = A os B si, where A = f d, 2 A = f os d, B = f si d Note: The formuls mke sese for piee-wise otiuous futios; i ft for ll solutely itegrle futios (ie f d ) Ay otiuous futio o [,] whih is periodi with period 2 e uiformly pproimted y trigoometri polyomils C N = C os D si More preisely, For ll there eists trigoometri polyomil p suh tht f p for ll [,], ie f p 2 There eists sequee of trigoometri polyomil p suh tht f p s Note: This does ot sy tht the Fourier series of f overges uiformly (or poitwise) to f Corollry The Fourier series of f overges to f i L 2 Note: This eed ot imply poitwise overgee Note: This remis true for pieewise otiuous futios Proof: Let { N } N = e the sequee of trigoometri polyomils whih overge uiformly to f N ivolves, 3 of 35

32 MAT337H Itrodutio to Rel Alysis os, d si for N Uiform overgee o fiite itervl implies L 2 overgee, hee N f i L 2 Let S N e the prtil sums of the Fourier series of f, ie S N =A A os B si By the Projetio, f S N 2 f N 2 ; i ft, f N 2 = f S N 2 S N N 2 Sie f N 2 s N, hee f S N 2 s N N = Corollry N f 2 d =A A B 2 Hee if the Fourier series of (pieewise) otiuous futio is, the the = futio is Therefore if f d g re (pieewise) otiuous futios with the sme Fourier series oeffiiets, the f g Proof: Follows from the Fourier series of f overgig to f i L 2 Propositio Suppose the Fourier series of otiuous periodi futio f overges uiformly to some futio The this futio mush e f Proof: Let g =A A os B si g is otiuous d periodi Wht re its Fourier oeffiiets? = [ A k= g d = A os k B si k ] os d =A fter iterhgig summtio d itegrtio Similrly for the other Fourier oeffiiets of g Therefore f g sie the hve the sme Fourier oeffiiets Lemm The Fourier oeffiiets of solutely itegrle futio re ouded y A f, A 2 f, B 2 f, Proof: A = 2 f d f d = f 2 If f is (pieewise) C f '~ = B os A si Proof: Let f '~ = futio whih is 2 -periodi d f ~A A os B si, the os si = f ' d = 2 2 [ f ] = sie f is periodi = f ' os d = [ f os ] = f si d = B = B 32 of 35

33 MAT337H Itrodutio to Rel Alysis = f ' si d = [ f si ] f os d = A = A If f is C 2 futio whih is 2 -periodi, the the Fourier series of f overges uiformly to f I prtiulr, it overges poitwise to f Proof: f ~A = A os B si, f '~ = B os A si, f ' ' ~ 2 A os 2 B si = f ' ' is otiuous o [,], hee f ' ' =M Hee 2 A M A M 2 d 2 B M B M 2 The the Weierstrss M-test, the series A Sie it overges uiformly, it must overge to f = A os B si overges uiformly Poitwise Covergee of Fourier Series If f stisfies Lipshitz oditio o [,], the the Fourier series of f overges poitwise to f Metri Spes DEFINITIONS AND EXAMPLES Defiitio: Metri Spe A metri spe is set X together with diste futio : X X [, suh tht, y,, y = = y ;, y = y, ;, z, y y, z (trigle iequlity) Remrk Ay ormed vetor spe is metri spe Ay suset of ormed vetor spe is metri spe where, y = y Emple The surfe of sphere i R 3 e mde ito metri spe ito two turl wys: Defie, y to e the stright lie diste (y tuelig through) 2 Defie, y to e the diste from to y log the irle joiig them Defiitio: Ope Bll Let X e metri spe A ope ll of eter d rdius r is B r ={ X, r } Defiitio: Ope Set A set A is ope if for ll A there eists r suh tht B r A 33 of 35

34 MAT337H Itrodutio to Rel Alysis Defiitio: Cotiuity A futio f : X R is otiuous t if for ll there eists suh tht f f wheever, More geerlly, if X d Y re metri spes d f : X Y, the f is otiuous t if for ll there eists suh tht Y f, f wheever X, f is otiuous if it is otiuous t ll X Defiitio: Covergee If { } = is sequee i X, we sy X if there eists N suh tht Y, wheever N Defiitio: Cuhy A sequee { } = i X is Cuhy if there eists M suh tht Y, m wheever,mn Defiitio: Completeess A metri spe X is omplete if y Cuhy sequee overges to poit i X Defiitio: Limit Poit Let A X d A is limit poit of A if there is sequee of poits { } = i A suh tht s Defiitio: Closed Set A suset A X is losed if it otis ll its limit poits A suset A X is ope if d oly if the omplemet of A is losed : Chrteriztios of Cotiuity Let X d Y e metri spes Let f : X Y e mppig The followig re equivlet: f is otiuous 2 Wheever { } = is sequee i X suh tht X, f f Y 3 Wheever U is ope suset of Y, f U is ope suset of X COMPACT METRIC SPACES Defiitio: Ope Cover A ope over of metri spe X is olletio of ope sets {U } B suh tht X B U Defiitio: Compt A metri spe X (or suspe A X ) is ompt if y ope over of X (or of A ) hs fiite suover Note: Suspe here do ot me vetor suspe Emple X =, is ot ompt Tke U =,, the U =, But if we tke fiitely my of these sets, sy =2 34 of 35

35 MAT337H Itrodutio to Rel Alysis U,U k d let N =m {,, k }, the U j U N j=,, k So U j =U N, j= k Emple X =[,] is ompt i this sese Defiitio: Sequetilly Compt A metri spe X is sequetilly ompt if every sequee { } = i X hs susequee { j } j= poit i X Note: I metri spe, omptess is equivlet to sequetil omptess whih overges to If f : X Y is otiuous mppig etwee metri spes X d Y d X is ompt, the f X is ompt Proof: Let {U } B e ope overig of f X The f U B is ope y otiuity, d { f U } B is ope overig of X Sie X is ompt, fiitely my of the f U, sy f U,, f U over X Hee U,,U over f X, ie f X j= U j This is true for ll ope overigs of f X, so f X is ompt 35 of 35