劍橋大學(xué)——數(shù)學(xué)分析.pdf

Analysis Dr J M E Hyland Mich lmas 1996 These notes are maintained by Paul Metcalfe Comments and corrections to soc archim notes lists cam ac uk Revision 1 6 Date 1999 09 17 17 34 31 The following people have maintained these notes initial typingRichard Cameron datePaul Metcalfe Contents Introductionv 1Real Numbers1 1 1Ordered Fields 1 1 2Convergence of Sequences 1 1 3Completeness of Bounded monotonic sequences 4 1 4Completeness of Least Upper Bound Principle 5 1 5Completeness of General Principle of Convergence 7 2Euclidean Space9 2 1The Euclidean Metric 9 2 2Sequences in Euclidean Space 9 2 3The Topology of Euclidean Space 11 2 4Continuity of Functions 12 2 5Uniform Continuity 15 3Differentiation17 3 1The Derivative 17 3 2Partial Derivatives 19 3 3The Chain Rule 20 3 4Mean Value Theorems 22 3 5Double Differentiation 23 3 6Mean Value Theorems in Many Variables 24 4Integration27 4 1The Riemann Integral 27 4 2Riemann s Condition A GPC for integrability 28 4 3Closure Properties 31 4 4The Fundamental Theorem of Calculus 33 4 5Differentiating Through the Integral 35 4 6Miscellaneous Topics 36 5Metric Spaces39 5 1 Defi nition and Examples 39 5 2Continuity and Uniform Continuity 41 5 3Limits of sequences 42 5 4Open and Closed Sets in Metric Spaces 44 5 5Compactness 45 5 6Completeness 47 iii ivCONTENTS 6Uniform Convergence49 6 1 Motivation and Defi nition 49 6 2The space 51 6 3The Integral as a Continuous Function 52 6 4Application to Power Series 54 6 5Application to Fourier Series 55 7The Contraction Mapping Theorem57 7 1Statement and Proof 57 7 2Application to Differential Equations 59 7 3Differential Equations pathologies 62 7 4Boundary Value Problems Green s functions 63 7 5The Inverse Function Theorem 65 Introduction These notes are based on the course Analysis given by Dr J M E Hyland in Cam bridge in the Mich lmas Term 1996 These typeset notes are totally unconnected with Dr Hyland 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 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 Real Numbers 1 1Ordered Fields Defi nition 1 1 A fi eld is a set equipped with an element and a binary operation making an abelian group we write for the additive inverse of an element and a binary operation such multiplication distributes over addition that is and multiplicationrestricts to and is anabeliangroup under multiplication we write for the multiplicative inverse of Examples rational numbers real numbers complex numbers Defi nition 1 2 A relation on a set is a strict total order when we have and or or for all and in We write for or and note that in a total order Familiar ordered fi elds are and but not 1 2Convergence of Sequences Defi nition 1 3 In an ordered fi eld we defi ne the absolute value of as and then we have the distance between and 1 2CHAPTER 1 REAL NUMBERS In an ordered fi eld the distance satisfi es and iff Proof Proof of this is easy Start from Add these to get Put for result In general the distance takes values in the fi eld in question but in the case of and the distance is real valued so we have a metric Example 1 4 Any ordered fi eld has a copy of as an ordered subfi eld Proof We set and so get and so get in all ordered correctly Defi nition 1 5 A sequence converges to a limit or tends to in an ordered fi eld just when for all in there exists with for all We write or as or just when converges to a limit So we have Example 1 6 1 iff 2 then 3 Suppose we have such that for all then iff 4 The sequence for does not converge 1 2 CONVERGENCE OF SEQUENCES3 Proof Suppose say Taking we can fi nd such that for all Then This is a contradiction and so does not converge 1 Lemma 1 7 Uniqueness of limit If and then Proof Given there exists such that implies and such that implies Let be the greater of and Now But is arbitrary so and Observation 1 8 Suppose and for all suffi ciently large Then Proof Suppose so that We can fi nd such that for all Consider So a contradiction We deduce Example 1 9 We know that in WHY There are ordered fi elds in which e g fi eld of rational functions ordered so that is infi nite Easy to see that in Proposition 1 10 Suppose that and Then 1 2 3 Proof of 1 and 2 are both trivial and are left to the reader Proof of 3 Given take such that for all and such that for all Let Now for all Now can be made arbitrarily small and the result is proved 1This is a rigorous form of the thought if we can t have both within of 4CHAPTER 1 REAL NUMBERS 1 3Completeness of Bounded monotonic sequences Defi nition 1 11 A sequence is monotonic increasing just when for all it is monotonic decreasing just when for all To cover either case we say the sequence is monotonic N B is increasing iff is decreasing A sequence is bounded above when there is with for all it is boundedbelow when there is with for all it is boundedwhen it is bounded above and below Axiom Completeness Axiom The real numbers form an ordered fi eld and every bounded monotonic sequence of reals has a limit ie converges Remarks This can be justifi ed on further conditions but here we take it as an axiom It is enough to say an increasing sequence bounded above converges In fact this characterizes as the completion of From now on we consider only the complete ordered fi eld and occasionally its incomplete ordered subfi eld Proposition 1 12 Archimedean Property 1 For any real there is with 2 For any there is with 3 The sequence Proof 1 Recall that is an increasing non convergent sequence Hence it is not bounded above and so for any there is with 2 If then consider and take with Then 3 Given we can fi nd with Now if and the result is proved Defi nition 1 13 If is a sequence and we have for with then is a subsequence of Observation 1 14 Suppose has a subsequence Then as 1 4 COMPLETENESS OF LEAST UPPER BOUND PRINCIPLE5 Theorem 1 15 The Bolzano Weierstrass Theorem Any bounded sequence of re als has a convergent subsequence Cheap proof Let be a bounded sequence Say that is a peak number iff for all Either there are infi nitely many peak numbers in which case we enumerate them in order Then and so is a bounded decreasing subsequence of so converges Or there are fi nitely many peak numbers Let be the greatest Then for every is not a peak number and so we can fi nd the least with Defi ne inductively by By defi nition for all and for all so is a bounded strictly increasing subsequence of and so converges This basis of this proof is that any sequence in a total order has a monotonic subse quence 1 4Completeness of Least Upper Bound Principle Defi nition 1 16 Let be a non empty set of reals is an upper bound for iff for all and if has such S is bounded above is a lower bound for iff for all and if has such S is bounded below is bounded iff is bounded above and below ie if for some is the least upper bound of or the supremum of iff is an upper bound If then for some ie is not an upper bound for Similarly is the greatest lower bound of or the infi mum of iff is a lower bound If then for some ie is not a lower bound 2 Notation Theorem 1 17 Least Upper Bound Principle A non empty set of reals which is bounded above has a least upper bound Proof Suppose and bounded above Take an upper bound and in say so that Set so that and defi ne inductively as follows Suppose given then are defi ned by stipulating 2Aside If are both least upper bounds of then can t have and can t have and so 6CHAPTER 1 REAL NUMBERS If then If otherwise then We can see inductively that 1 for all 2 for all 3 for all 3 4 is an upper bound of for every 4 By 1 is decreasing bounded below by so say is increasing bounded above by so By 2 as But and so Claim is Each is an upper bound of and so is an upper bound for if we have all and so Take We can take such that for all 5 But then and so there is such that This shows that is the least upper bound Observation 1 18 We can deduce the completeness axiom from the LUB principle Proof If is increasing and bounded above then is non empty and bounded above and so we can set Suppose given Now and so there is with but then for and so 3True for and inductively certainly true for in fi rst alternative and in the 2nd alternative since by induction hypothesis 4True for and inductively trivial in fi rst case and in the second clear as 5Let We can fi nd such that and thus 1 5 COMPLETENESS OF GENERAL PRINCIPLE OF CONVERGENCE7 1 5Completeness of General Principle of Conver gence Defi nition 1 19 A real sequence is a Cauchy Sequence if and only if for all there exists with That is is Cauchy iff Observation 1 20 A Cauchy sequence is bounded For if is Cauchy take such that for all Then is bounded by Lemma 1 21 Suppose is Cauchy and has a convergent subsequence as Then as Proof Given take such that for all and take with easy enough to require such that for all Then if But can be made arbitrarily small so Theorem 1 22 The General Principle of Convergence A real sequence converges if and only if it is Cauchy Proof Suppose Given take such that for all Then if As can be made arbitrarily small is Cauchy Suppose is Cauchy 6 Then is bounded and so we can apply Bolzano Weierstrass to obtain a convergent subsequence as By lemma 1 21 Alternative Proof Suppose is Cauchy Then it is bounded say Consider for infi nitely many First and so is bounded above by in fact by By the LUB principle we can take 6This second direction contains the completeness information 8CHAPTER 1 REAL NUMBERS Given andsothereis with Thenthereareinfi nitely many with so and so there are only fi nitely many with Thus there are infi nitely many with Take such that for all We can fi nd with ie Then if As can be made arbitrarily small this shows Remarks This second proof can be modifi ed to give a proof of Bolzano Weierstrass from the LUB principle In the proof by bisection of the LUB principle we could have used GPC general principle of convergence instead of Completeness Axiom We can prove GPC directly from completeness axiom as follows Given Cauchy defi ne is increasing so Then show The Completeness Axiom LUB principle and the GPC are equivalent expres sions of the completeness of Chapter 2 Euclidean Space 2 1The Euclidean Metric Recallthat isavectorspacewithcoordinate wiseadditionandscalarmultiplication Defi nition 2 1 The Euclidean norm1 is defi ned by and the Euclidean distance between and is Observation 2 2 The norm satisfi es and the distance satisfi es 2 2Sequences in Euclidean Space We can write or for a sequence of points in Then for the thcoordinate of the thnumber of the sequence 1The norm arises from the standard inner product 9 10CHAPTER 2 EUCLIDEAN SPACE Defi nition 2 3 A sequence converges to in when for any there exists such that 2 for all In symbols Proposition 2 4 in iff in for Proof Note that and Defi nition 2 5 A sequence is boundedif and onlyif there exists such that for all Theorem 2 6 Bolzano Weierstrass Theorem for Any bounded sequence in has a convergent subsequence Proof Version 1 Suppose is bounded by Then all the coordinates are bounded by By Bolzano Weierstrass in we can take a subsequence such that the 1st coordinates converge now by Bolzano Weierstrass we can take a subsequence of this sequence such that the 2nd coordinates converge Continuing in this way in steps we get a subsequenceall of whose coordinatesconverge But then this converges in Version 2 By induction on The result is known for Bolzano Weierstrass in and is trivial for Suppose result is true for Take a bounded subsequence in and write each as where and is the thcoordinate Now and are both bounded so we can apply Bolzano Weierstrass in to get a subsequence Apply Bolzano Weierstrass in to get Then as 2 in iff in 2 3 THE TOPOLOGY OF EUCLIDEAN SPACE11 Defi nition 2 7 A sequence is a Cauchy sequence iff for any there is with for In symbols this is Observation 2 8 is Cauchy in iff each is Cauchy in for Proof Suppose is Cauchy Take Given we can fi nd such that for all But then for so as is arbitrary is Cauchy Conversely suppose each is Cauchy for Given we can fi nd such that Now if then As can be made arbitrarily small is Cauchy Theorem 2 9 General Principle of Convergence in A sequence in is convergent if and only if is Cauchy Proof converges in iff converges in iff is Cauchy in iff is Cauchy in 2 3The Topology of Euclidean Space For and we have the open ball defi ned by Also we have the closed ball defi ned by Also we shall sometimes need the punctured open ball Defi nition 2 10 A subset is open if and only if for all there exists such that That is is open iff for all there exists with 12CHAPTER 2 EUCLIDEAN SPACE The empty set is trivially open Example 2 11 is open for if then setting we see Similarly is open But is not open for any Defi nition 2 12 A subset is closed iff whenever is a sequence in and then In symbols this is Example 2 13 is closed for suppose and then for all Now As and so as Therefore A product of closed intervals is closed For if and then each with so that Therefore But is not closed unless Proposition 2 14 A set is open in iff its complement is closed in A set is closed in iff its complement is open in 3 Proof Exercise 2 4Continuity of Functions We consider functions defi ned on some For now imagine that is a simple open or closed set as in 2 3 3Warning Sets need not be either open or closed the half open interval is neither open nor closed in 2 4 CONTINUITY OF FUNCTIONS13 Defi nition 2 15 Suppose with Then is continuous at iff for any there exists 4 such that for all In symbols is continuous iff is continuous at every point This can be reformulated in terms of limit notation as follows Defi nition 2 16 Suppose Then as in 5if an only if for any there exists such that for all Remarks We typically use this when is open and some punctured ball is contained in Then the limit notion is independent of If as thendefi ning extends to a functioncontinuous at Proposition 2 17 Suppose is continuous in if and only if whenever in then This is known as sequential continuity is continuous in if and only if for any open subset is open in Proof We will only prove the fi rst part for now The proof of the second part is given in theorem 5 16 in a more general form Assume is continuousat andtakea convergentsequence in Suppose given By continuity of there exists such that As take such that for all Now if Since can be made arbitrarily small The converse is clear 4The continuity of at depends only on the behavior of in an open ball 5Then is continuous at iff as in 14CHAPTER 2 EUCLIDEAN SPACE Remark as iff as Observation 2 18 Any linear map is continuous Proof If has matrix with respect to the standard basis then and so Fix Given we note that if then As can be made arbitrarily small is continuous at But arbitrary so is continuous If is continuous at and is continuous at then is continuous at Proof Given take such that Take such that Then Proposition 2 19 Suppose are continuous at Then 1 is continuous at 2 is continuous at any 3 If is continuous at Proof Proof is trivial Just apply propositions 1 10 and 2 17 Suppose Then we can write where is composed with the thprojection or coordinate function Then is continuous if and only if each is continuous 2 5 UNIFORM CONTINUITY15 Theorem 2 20 Suppose that is continuous on a closed and bounded subset of Then is bounded and so long as attains its bounds Proof Suppose not bounded Then we can take with By Bolzano Weierstrass we can take a convergent subsequence as and as is closed By the continuity of as But is unbounded a contradiction and so is bounded Now suppose We can take with By Bolzano Weierstrass we can take a convergentsubsequence As is closed By continuityof but by construction as So Essentially the same argument shows the more general fact If is continuous in c