剑桥大学——高等数学分析.pdf

Further Analysis Prof W T Gowers Lent 1997 These notes are maintained by Paul Metcalfe Comments and corrections to soc archim notes lists cam ac uk Revision 2 8 Date 1999 10 22 11 33 59 The following people have maintained these notes datePaul Metcalfe Contents Introductionv 1Topological Spaces1 1 1Introduction 1 1 2Building New Spaces 2 2Compactness5 2 1Introduction 5 2 2Some compact sets 5 2 3Consequences of compactness 7 2 4Other forms of compactness 7 3Connectedness9 3 1Introduction 9 3 2Connectedness inR 10 3 3Path connectedness 10 4Preliminaries to complex analysis13 4 1Paths 13 4 2Complex Integration 13 4 3Domains 14 4 4Path Integrals 15 5Cauchy s theorem and its consequences19 5 1Cauchy s theorem 19 5 2Homotopy 20 5 3Consequences of Cauchy s Theorem 23 6Power Series27 6 1Analyticity and Holomorphy 27 6 2 Classifi cation of Isolated Singularities 31 7Winding Numbers35 7 1 Introduction and Defi nition 35 7 2Residues 37 8Cauchy s Theorem homology version 41 iii ivCONTENTS Introduction These notes are based on the course Further Analysis given by Prof W T Gowers1 in Cambridge in the Lent Term 1997 These typeset notes are totally unconnectedwith Prof Gowers Other sets of notes are available for different courses At the time of typing these courses were ProbabilityDiscrete Mathematics AnalysisFurther Analysis MethodsQuantum Mechanics Fluid Dynamics 1Quadratic Mathematics GeometryDynamics of D E s Foundations of QMElectrodynamics Methods of Math PhysFluid Dynamics 2 Waves etc Statistical Physics General RelativityDynamical Systems Physiological Fluid DynamicsBifurcations in Nonlinear Convection Slow Viscous FlowsTurbulence and Self Similarity AcousticsNon Newtonian Fluids Seismic Waves They may be downloaded from http www istari ucam org maths or http www cam ac uk CambUniv Societies archim notes htm or you can email soc archim notes lists cam ac uk to get a copy of the sets you require 1Yes that Prof Gowers v Copyright c The Archimedeans Cambridge University All rights reserved Redistribution and use of these notes in electronic or printed form with or without modifi cation are permitted provided that the following conditions are met 1 Redistributions of the electronic fi les must retain the abovecopyrightnotice this list of conditions and the following disclaimer 2 Redistributions in printed form must reproduce the above copyright notice this list of conditions and the following disclaimer 3 All materials derived from these notes must display the following acknowledge ment ThisproductincludesnotesdevelopedbyTheArchimedeans Cambridge University and their contributors 4 Neither the name of The Archimedeans nor the names of their contributors may be used to endorse or promote products derived from these notes 5 Neither these notes nor any derived products may be sold on a for profi t basis although a fee may be required for the physical act of copying 6 You must cause any edited versions to carry prominent notices stating that you edited them and the date of any change THESENOTESAREPROVIDEDBYTHEARCHIMEDEANSANDCONTRIB UTORS AS IS ANDANYEXPRESSORIMPLIEDWARRANTIES INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABIL ITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED IN NO EVENT SHALL THE ARCHIMEDEANS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT INDIRECT INCIDENTAL SPECIAL EXEMPLARY OR CONSE QUENTIAL DAMAGES HOWEVER CAUSED AND ON ANY THEORY OF LI ABILITY WHETHER IN CONTRACT STRICT LIABILITY OR TORT INCLUD ING NEGLIGENCE OR OTHERWISE ARISING IN ANY WAY OUT OF THE USE OF THESE NOTES EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAM AGE Chapter 1 Topological Spaces 1 1Introduction Defi nition 1 1 A topological space is a setXtogether with a collection of subsets ofXsatisfying the following axioms 1 X 2 IfU U n thenU U n that is is closed under fi nite intersections 3 Any union of sets in is in or is closed under any unions Defi nition 1 2 is called a topology onX The sets in are called open sets A subset ofXis closed if its complement is open Examples 1 If X d is a metric space and the collection of open sets in a metric space sense then X is a topological space 2 IfXis any set and is the power set ofX X is a topological space is called the discrete topology onX 3 IfXis any set and f Xg X is a topological space is called the indiscrete topology onX 4 IfX is any infi nite set and fY X XnY is fi niteg f gthen X is a topological space is called the cofi nite topology onX 5 IfXis any uncountableset and fY X XnYis countableg f gthen X is a topological space is called the cocountable topology onX Defi nition 1 3 LetAbe a subset of a topological space The closure ofA denotedA is the intersection of all closed sets containingA Note thatAis closed and any closed set containingAcontainsA Defi nition 1 4 LetAbe a subset of a topological space The interior ofA denoted intAorA is the union of all open sets inA Note thatintAis open and any open set inAis inintA Defi nition 1 5 The boundary Aof a set A isAnintA 1 2CHAPTER 1 TOPOLOGICAL SPACES Defi nition 1 6 Let x n be a sequence in a topological space X We say that x nconverges to x x n x if for every open setUsuch thatx U Nsuch that n N x n U This agrees with the usual defi nition for metric spaces Defi nition 1 7 Let X and Y be topological spaces and letf X Y We say thatfis continuousif for everyU f U Or inverse image of an open set is open It follows from results in Analysis that this defi nition agrees with the usual defi nition ifXandYare metric spaces Defi nition 1 8 Let X be a topological space and letx X A neighbourhood of xis a setNthat contains an open set containingx Proposition 1 9 Let X and Y be topological spaces and letf X Y Then the following are equivalent 1 fis continuous 2 For everyx Xand every neighbourhoodMoff x there exists a neighbour hoodNofxsuch thatf N M Proof Firstly do Mcontains an open setUcontainingf x ThenN f U is a neighbourhoodofxsuch thatf N U Now dothe case LetUbean opensubset ofY Foreveryx f U Uis a neighbourhoodoff x We can fi nd a neighbourhoodN xof xsuch thatf N x U NowN xcontains an open set V xcontaining x Let V x f U V x Thenf V x U x sof V UandV f U ButV f U soV f U Vis open sof U is open andfis continuous Defi nition 1 10 Let X and Y be topological spaces andf X Y fis a homeomorphism if it is continuous with a continuous inverse Example Rand with usual metrics are homeomorphic 1 2Building New Spaces Defi nition 1 11 Let X be a topological space andY X The subspace topol ogy onYis fU Y U g Proposition 1 12 Let X d be a metric space Y XandU Y Then the following are equivalent 1 Uis open in the subspace topology onY 2 For everyu U such that ifv Y d u v thenv U 1 2 BUILDING NEW SPACES3 Proof Do Letu U SinceUopen inY V X Vopen such that U V Y Then such thatd u v v V Nowd u v and v Y v V Y U Now do SupposeU satisfi es 2 For everyu U pick u such thatd u v andv Y v U LetN u fv X d u v g NowN uis open andV S u U N uis open Now V Y U Defi nition 1 13 Let X be a topological space and be an equivalence relation onX Denote the setX byYand letq X Ybe the equivalence map ie if x X q x is the equivalence class ofx The quotient topology onYisfU Y q U g N B 1 q U is the union of the equivalence classes in U 2 Noticeqis continuous and that the quotient topology is the largest topology making it so Examples 1 Let be the usual topology onR Then the subspace topology on Z Rcoincides with the discrete topology onZ 2 Let be as in 1 Then the interval is open in the subspace topology on 3 Let be the equivalence relation onR defi ned byx y x y Z Let x denote the equivalence class ofx Then the map R T fz C jzj g is both well defi ned and a homeomorphism Defi nition 1 14 Let X and Y be topological spaces The product topology onX Yis the collection of all possible unions of sets of the formU VwithU andV Similarly if X i i n i are ntopological spaces the product topology on Q n i X iis the collection of all unions of sets Q n i U iwith U i i Example IfRhas its usual topology then the product topology onR Ris the same as the usual topology onR R R Proof Let us write for the product topology onR Rand for the usual Eu clidean topology LetU Then given x x U such that d x x y y y y U Then the open set x x x x Uand contains x x soUis open given Conversely every set of the formA BwithA Bopen inRis open inR The union of such sets is open giving and Exercise to show thatX Y Z X Y Zas topological spaces Defi nition 1 15 Let X be a topological space A basis for or a basis of open sets is a subset such that everyU is a union of sets in The sets in are called basic open sets Ifx X then a basis of neighbourhoods ofxis a collection Nof neighbourhoodsofxsuch that every neighbourhoodofxcontainsN N Examples The setsU VwithU andV are a basis for the product topology on X Y 4CHAPTER 1 TOPOLOGICAL SPACES The setsfy d x y n gare a basis of neighbourhoods for a pointxin a metric space Proposition 1 16 The quotient topology and product topology are topologies Proof The result for the quotient topology follows easily from the fact thatq pre serves unions and intersections For products let X i u n i be topological spaces Everything is simple except closure under fi nite intersections By induction it is suffi cient to prove for two sets IfU i i call U U nan open box1 Firstly observe that U U n V V n U V U n V n and thus the intersection of two open boxes is an open box Now take B and C with allB s and C s open boxes Now B C B C which is a union of open boxes and so open 1This is not standard terminology Chapter 2 Compactness 2 1Introduction Defi nition 2 1 Let X be a topological space An open cover ofXis a collection fU gof open sets such thatX S U If Y Xthen an open cover of Yis a collectionfU gsuch thatY S U A subcover of a coverU fU gis a subsetV Uwhich is still an open cover Examples 1 fI n n n n gis an open cover forR fI n gis a subcover 2 The intervalsI n n n withn Zform a cover of the reals with no proper subcover Defi nition 2 2 A topological space X is compact if every open cover has a fi nite subcover Examples The open covers mentioned above show thatR is not compact Any fi nite topological space is compact as is any set with the indiscrete topology 2 2Some compact sets Lemma 2 3 Let X be a topological space withY X Then the following are equivalent 1 Yis compact in the subspace topology 2 Every cover ofYbyU has a fi nite subcover Proof LetY S U with U ThenY S U Y withU open inY SinceYis compact nsuch that Y S n i U i Y which gives thatY S n i U i LetY S V with V open in Y SinceV U Yfor someU open in X Y S U By assumption nsuch that Y S n i U i This implies thatY S n i U i Y S n i V i 5 6CHAPTER 2 COMPACTNESS Theorem 2 4 The Heine Borel Theorem The closed interval a b Ris com pact Proof Let a b S U with U open in Rand let K fx a b a x is contained in a fi nite union of open setsg Letr supK Thenr a b sor U for some ButU is open so such that r r U By the defi nition ofr a r has a fi nite open cover thus a r has a fi nite open cover This is a contradiction unlessr b Theorem 2 5 A continuous image of a compact set is compact Proof LetXbe compact and letf X Ybe continuous Nowf X S U withU open in Y Sincefis continuous f U is open inX and furthermoreX S f U SinceXis compact nsuch that X S n i f U i Thusf X S n i U i Theorem 2 6 A closed subset of a compact set is compact Proof LetXbe compact andK Xbe closed LetK S U with U open in X NowX XnK S U SinceXis compact a fi nite subcover and X XnK S n i U i ThenK S n i U i Defi nition 2 7 A topological space X is called Hausdorff if given any two dis tinctx y X there exist disjoint open setsUandVwithx Uandy V Examples 1 Any metric space is Hausdorff Givenx ydistinct letU fz d x z d x y gandV fz d y z d x y g 2 Any set with more than one element and the indiscrete topology is not Hausdorff Theorem 2 8 Every compact subset of a Hausdorff topological space is closed Proof LetXbe HausdorffandK Xbe compact Ifx Kandy Kthen one can fi nd disjoint open setsU xyand V xywith x U xyand y V xy For fi xedx the sets V xy y Kform an open cover ofK Hence y y nsuch that K S n i V xy i LetU x T n i U xy iand V x S n i V xy i Note that U x V x x U x K V x andU x K But S x K U x XnKis open Theorem 2 9 A product of fi nitely many compact sets is compact Proof It is enough to do two sets The general result follows by induction andX Y Z X Y Z LetXandYbe compact and letX Y S U U open in X Y U is the union of sets of the formV W withVopen inXandWopen inY It follows thatX Y S V W with V open in XandW open in Y V W U Letx X Now fxg Y x V V W SinceY is compact one can fi nd nsuch that Y S n i W i Let V x T n i V i The sets V xform an open cover of X thus x x nsuch than X 2 3 CONSEQUENCES OF COMPACTNESS7 S m j V x j Now X Y S m j V x j Y ButV x j Y has a fi nitecoverof V W s For eachV W choose U with V W U to obtain a fi nite open cover of X Y Theorem 2 10 A subset ofR n is compact iff it is closed and bounded Proof LetX R n be closed and bounded Then Msuch thatd x M x X In particular X M M n But M M is compact so M M n is com pact ButXis closed soXis compact Conversely ifXis compact thenXis closed sinceR n is Hausdorff IfXis not bounded then the setsU m fx X d x mgform an open cover with no fi nite subcover 2 3Consequences of compactness Theorem 2 11 A continuous real function on a compact metric space is bounded and attains its bounds Proof The image of such a function is compact and so closed and bounded Closed the function attains its bounds Theorem 2 12 LetXbe a compact metric space letYbe a metric space and let f X Ybe continuous Thenfis uniformly continuous Proof Let be arbitrary We must show x y Xd x y d f x f y Sincefis continuous x X x y Xd x y x d f x f y Now letU x fy d x y x g Then theU xform an open cover of X so x x nsuch that X S n i U x i Let min x i Letd y z Since theU x i form a cover we can fi ndisuch thatd x i y x i andd x i z x i By the defi nition of theU x i d f y f z by the triangle inequality Lemma 2 13 LetXbe a metric space and letK Xbe compact andY Xbe closed Then x Ksuch thatd x Y inffd x w x K w Yg Proof Defi nef K Rbyf x dist x Y This is continuous and so attains its lower bound In particular ifKandYare disjoint thend x Y for everyxasYis closed Hence such thatd x Y x X 2 4Other forms of compactness Defi nition 2 14 Xis sequentially compact if every sequence inXhas a convergent subsequence Bolzano Weierstrass is the statement that a closed boundedsubset ofR n is sequen tially compact Theorem 2 15 A compact metric space is sequentially compact 8CHAPTER 2 COMPACTNESS Proof LetXbe a metric space and let x n be a sequence with no convergent sub sequence Now claim that x X such thatd x x n for at most fi nitely manyn Otherwise xsuch thatd x x n m infi nitely many times Now easy to construct a convergentsubsequence For eachx pick such a and letU x fy d x y g TheU xthus form an open cover with no fi nite subcover Theorem 2 16 LetX R n Then the following are equivalent 1 Xis compact 2 Xis sequentially compact 3 Xis closed and bounded Proof We know from previous theorems Thus enough to show that IfXis not closed x Xsuch that m y m Xsuch thatd y m x n Then every subsequence ofy nconverges in R n toxand does not converge inX IfX is not bounded then n x nsuch that d x n n a sequence with no convergent subsequence Chapter 3 Connectedness 3 1Introduction Defi nition 3 1 LetXbe a topological space SupposeU V Xsuch that 1 U Vopen 2 U V 3 X U V 4 bothUandVare non empty ThenU Vare said to disconnectX Xis connected if no two subsets disconnectX IfYis a subspace of a topologicalspaceX thenYis disconnectedin the subspace topology iff U Vopen inXsuch thatU Y V Y andY U Vand Y U V In this case we shall say thatU VdisconnectY Proposition 3 2 LetXbe a topological space Then the following are equivalent 1 Xis connected 2 Every continuousf X Zis constant 3 The only subsets ofXboth open and closed are andX Proof Suppose non constantf X Z Then we can fi ndm nsuch that bothmandnare inf X Thenf fk k mg andf fk k mg disconnectX SupposeUandVdisconnectX Then consider f x x U x V Note that ifU X U XnU X Proposition 3 3 A continuous image of a connected space is connected Proof Letf X Ybe continuous withXa connected topological space Then if UandVdisconnectf X f U andf V disconnectX 9 10CHAPTER 3 CONNECTEDNESS 3 2Connectedness inR Defi nition 3 4 A subsetIof the reals is an interval if wheneverx y z x z I y I Every interval is of one of these nine forms a b a b a a b a b a b b To see this note that at the upper end of the interval there are three possibilities bounded and achieves bound bounded and does not achieve bound and unbounded Similarly for the lower end of the interval Theorem 3 5 A subset ofRis connected iff it is an interval Proof IfX R is not an interval we can fi ndx y zsuch thatx z Xand y X Then y and y disconnectX Now letI Rbe an interval and supposeUandVdisconnectI We can fi nd u U Iandv V Iand without loss of generality takeu v SinceIis an interval u v I Lets supf u v Ug Ifs Uthens v SinceUis open such that s s U Ifs Vthen s s V s s U Corollary 3 6 The Intermediate Value Theorem Leta bandf a b Rbe continuous Iff a y f b then x a b such thatf x y Proof If not f y andf y disconnect a b 3 3Path connectedness Defi nition 3 7 LetXbe a topological space and letx y X A continuous path fromxtoyis a continuous function a b Xsuch that a xand b y Defi nition 3 8 Xis path connected if x y X a path fromxtoy Proposition 3 9 Path connectedness implies connectedness Proof SupposeXis a topological space andUandVdisconnectX Letu Uand v V IfXis path connected then continuous a b Xwith a uand b v Then U and V disconnect a b Defi nition 3 10 LetXbe a topological space and a b Xand c d X be continuous paths with b c Then the join of and written is defi ned as a b d c X t t a t b t c b b t b d c Defi nition 3 11 The reverse of written is the path b a Xwith t t Let us writex yif a continuouspath inXfromxtoy This is an equivalence relation Writingx yfor is a path fromxtoy we have that 1 x y y x 3 3 PATH CONNECTEDNESS11