ANALYSIS TOOLS WITH APPLICATIONS

ANALYSIS TOOLS WITH APPLICATIONS

BRUCE K. DRIVER

Date: April 10, 2003 File:anal.tex . Department of Mathematics, 0112. University of California, San Diego . La Jolla, CA 92093-0112 .

i

ii

Abstract. These are lecture notes from Math 240. Things to do: 0) Exhibit a non-measurable null set and a non-Borel measurable Riemann

integrable function. 1) Weak convergence on metric spaces. See Durrett, Stochastic calculus,

Chapter 8 for example. Also see Stroock's book on this point, chapter 3. See Problems 3.1.18--3.1.20.

2) Infinite product measures using the Caratheodory extension theorem in the general case of products of arbitrary probability spaces. See Stroock's book on probability from an analytic point of view.

3) Do enough on topological vector spaces to cover what is needed for the section on distributions, this includes Banach - Steinhauss theorem and open mapping theorem in the context of Frechet spaces. See Rudin's functional analysis and len's notes.

4) Add manifolds basics including Stoke's theorems and partitions of unity. See file Partitn.tex in 257af94 directory. Also add facts about smooth measure on manifolds, see the last chapter of bookall.tex for this material.

5) Also basic ODE facts, i.e. flows of vector fields 6) Put in some complex variables. 7) Bochner Integrals (See Gaussian.tex for a discussion and problems below.) 8) Add in implicit function theorem proof of existence to ODE's via Joel Robbin's method, see PDE notes. 9) Manifold theory including Sards theorem (See p.538 of Taylor Volume I and references), Stokes Theorem, perhaps a little PDE on manifolds. 10) Put in more PDE stu, especially by hilbert space methods. See file zpde.tex in this directory. 11) Add some functional analysis, including the spectral theorem. See Taylor volume 2. 12) Perhaps some probability theory including stochastic integration. See course.tex from 257af94 and other files on disk. For Kolmogorov continuity criteria see course.tex from 257af94 as well. Also see Gaussian.tex in 289aW98 for construction of Wiener measures. 13) There are some typed notes on Partitions of unity called partitn.tex, from PDE course and other notes from that course may be useful. For more ODE stu see pdenote2.tex from directory 231a-f96. These notes also contain quadratic form notes and compact and Fredholm operator notes. 15) Move Holder spaces much earlier in the text as illustrations of compactness theorems. 14) Use the proof in Loomis of Tychono 's theorem, see p.11 15) Perhaps the pi-lambda theorem should go in section 4 when discussing the generation of -- algebras. Major Break down thoughts: I Real Analysis II: Topology III: Complex Variables IV Distributrion Theory, PDE 1 V: Functional analysis and PDE 2. (Sobolev Spaces) VI: Probability Theory VII: Manifold Theory and PDE 3.

Contents

1. Introduction

1

ANALYSIS TOOLS W ITH APPLICATIONS

iii

2. Limits, sums, and other basics

1

2.1. Set Operations

1

2.2. Limits, Limsups, and Liminfs

2

2.3. Sums of positive functions

3

2.4. Sums of complex functions

6

2.5. Iterated sums

9

2.6. cs -- spaces, Minkowski and Holder Inequalities

11

2.7. Exercises

15

3. Metric, Banach and Topological Spaces

18

3.1. Basic metric space notions

18

3.2. Continuity

20

3.3. Basic Topological Notions

21

3.4. Completeness

27

3.5. Compactness in Metric Spaces

29

3.6. Compactness in Function Spaces

34

3.7. Bounded Linear Operators Basics

36

3.8. Inverting Elements in O([) and Linear ODE

40

3.9. Supplement: Sums in Banach Spaces

42

3.10. Word of Caution

43

3.11. Exercises

45

4. The Riemann Integral

48

4.1. The Fundamental Theorem of Calculus

51

4.2. Exercises

53

5. Ordinary Dierential Equations in a Banach Space

55

5.1. Examples

55

5.2. Linear Ordinary Dierential Equations

57

5.3. Uniqueness Theorem and Continuous Dependence on Initial Data 60

5.4. Local Existence (Non-Linear ODE)

61

5.5. Global Properties

63

5.6. Semi-Group Properties of time independent flows

68

5.7. Exercises

70

6. Algebras, -- Algebras and Measurability

75

6.1. Introduction: What are measures and why "measurable" sets

75

6.2. The problem with Lebesgue "measure"

76

6.3. Algebras and -- algebras

78

6.4. Continuous and Measurable Functions

84

6.5. Topologies and -- Algebras Generated by Functions

87

6.6. Product Spaces

89

6.7. Exercises

95

7. Measures and Integration

97

7.1. Example of Measures

99

7.2. Integrals of Simple functions

101

7.3. Integrals of positive functions

103

7.4. Integrals of Complex Valued Functions

110

7.5. Measurability on Complete Measure Spaces

117

7.6. Comparison of the Lebesgue and the Riemann Integral

118

7.7. Appendix: Bochner Integral

121

7.8. Bochner Integrals

124

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BRUCE K. DRIVER

7.9. Exercises

126

8. Fubini's Theorem

129

8.1. Measure Theoretic Arguments

129

8.2. Fubini-Tonelli's Theorem and Product Measure

135

8.3. Lebesgue measure on Rg

141

8.4. Polar Coordinates and Surface Measure

144

8.5. Regularity of Measures

147

8.6. Exercises

151

9. Os-spaces

153

9.1. Jensen's Inequality

156

9.2. Modes of Convergence

159

9.3. Completeness of Os -- spaces

162

9.4. Converse of H?lder's Inequality

166

9.5. Uniform Integrability

171

9.6. Exercises

176

10. Locally Compact Hausdor Spaces

178

10.1. Locally compact form of Urysohn Metrization Theorem

184

10.2. Partitions of Unity

186

10.3. F0([) and the Alexanderov Compactification

190

10.4. More on Separation Axioms: Normal Spaces

191

10.5. Exercises

194

11. Approximation Theorems and Convolutions

197

11.1. Convolution and Young's Inequalities

201

11.2. Classical Weierstrass Approximation Theorem

208

11.3. Stone-Weierstrass Theorem

213

11.4. Locally Compact Version of Stone-Weierstrass Theorem

216

11.5. Dynkin's Multiplicative System Theorem

217

11.6. Exercises

218

12. Hilbert Spaces

222

12.1. Hilbert Spaces Basics

222

12.2. Hilbert Space Basis

230

12.3. Fourier Series Considerations

232

12.4. Weak Convergence

235

12.5. Supplement 1: Converse of the Parallelogram Law

238

12.6. Supplement 2. Non-complete inner product spaces

240

12.7. Supplement 3: Conditional Expectation

241

12.8. Exercises

244

12.9. Fourier Series Exercises

246

12.10. Dirichlet Problems on G

250

13. Construction of Measures

253

13.1. Finitely Additive Measures and Associated Integrals

253

13.2. The Daniell-Stone Construction Theorem

257

13.3. Extensions of premeasures to measures I

261

13.4. Riesz Representation Theorem

263

13.5. Metric space regularity results resisted

269

13.6. Measure on Products of Metric spaces

270

13.7. Measures on general infinite product spaces

272

13.8. Extensions of premeasures to measures II

274

ANALYSIS TOOLS W ITH APPLICATIONS

v

13.9. Supplement: Generalizations of Theorem 13.35 to Rq

277

13.10. Exercises

279

14. Daniell Integral Proofs

282

14.1. Extension of Integrals

282

14.2. The Structure of O1(L)

289

14.3. Relationship to Measure Theory

290

15. Complex Measures, Radon-Nikodym Theorem and the Dual of Os 296

15.1. Radon-Nikodym Theorem I

297

15.2. Signed Measures

302

15.3. Complex Measures II

307

15.4. Absolute Continuity on an Algebra

310

15.5. Dual Spaces and the Complex Riesz Theorem

312

15.6. Exercises

314

16. Lebesgue Dierentiation and the Fundamental Theorem of Calculus 316

16.1. A Covering Lemma and Averaging Operators

316

16.2. Maximal Functions

317

16.3. Lebesque Set

319

16.4. The Fundamental Theorem of Calculus

322

16.5. Alternative method to the Fundamental Theorem of Calculus

330

16.6. Examples:

332

16.7. Exercises

333

17. More Point Set Topology

335

17.1. Connectedness

335

17.2. Product Spaces

337

17.3. Tychono's Theorem

339

17.4. Baire Category Theorem

341

17.5. Baire Category Theorem

341

17.6. Exercises

346

18. Banach Spaces II

348

18.1. Applications to Fourier Series

353

18.2. Hahn Banach Theorem

355

18.3. Weak and Strong Topologies

359

18.4. Weak Convergence Results

360

18.5. Supplement: Quotient spaces, adjoints, and more reflexivity

364

18.6. Exercises

368

19. Weak and Strong Derivatives

371

19.1. Basic Definitions and Properties

371

19.2. The connection of Weak and pointwise derivatives

382

19.3. Exercises

387

20. Fourier Transform

389

20.1. Fourier Transform

390

20.2. Schwartz Test Functions

392

20.3. Fourier Inversion Formula

394

20.4. Summary of Basic Properties of F and F1

397

20.5. Fourier Transforms of Measures and Bochner's Theorem

397

20.6. Supplement: Heisenberg Uncertainty Principle

400

21. Constant Coe!cient partial dierential equations

405

21.1. Elliptic Regularity

416

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BRUCE K. DRIVER

21.2. Exercises

420

22. O2 -- Sobolev spaces on Rq

421

22.1. Sobolev Spaces

421

22.2. Examples

429

22.3. Summary of operations on K4

431

22.4. Application to Dierential Equations

433

23. Sobolev Spaces

436

23.1. Mollifications

437

23.2. Dierence quotients

442

23.3. Application to regularity

443

23.4. Sobolev Spaces on Compact Manifolds

444

23.5. Trace Theorems

447

23.6. Extension Theorems

451

23.7. Exercises

453

24. H?lder Spaces

455

24.1. Exercises

460

25. Sobolev Inequalities

461

25.1. Gagliardo-Nirenberg-Sobolev Inequality

461

25.2. Morrey's Inequality

465

25.3. Rademacher's Theorem

470

25.4. Sobolev Embedding Theorems Summary

470

25.5. Other Theorems along these lines

471

25.6. Exercises

472

26. Banach Spaces III: Calculus

473

26.1. The Dierential

473

26.2. Product and Chain Rules

474

26.3. Partial Derivatives

476

26.4. Smooth Dependence of ODE's on Initial Conditions

477

26.5. Higher Order Derivatives

479

26.6. Contraction Mapping Principle

482

26.7. Inverse and Implicit Function Theorems

484

26.8. More on the Inverse Function Theorem

487

26.9. Applications

490

26.10. Exercises

492

27. Proof of the Change of Variable Theorem

494

27.1. Appendix: Other Approaches to proving Theorem 27.1

498

27.2. Sard's Theorem

499

27.3. Co-Area Formula

503

27.4. Stokes Theorem

503

28. Complex Dierentiable Functions

504

28.1. Basic Facts About Complex Numbers

504

28.2. The complex derivative

504

28.3. Contour integrals

509

28.4. Weak characterizations of K( )

515

28.5. Summary of Results

519

28.6. Exercises

520

28.7. Problems from Rudin

522

29. Littlewood Payley Theory

523

ANALYSIS TOOLS W ITH APPLICATIONS

vii

30. Elementary Distribution Theory

529

30.1. Distributions on X r Rq

529

30.2. Other classes of test functions

536

30.3. Compactly supported distributions

541

30.4. Tempered Distributions and the Fourier Transform

543

30.5. Appendix: Topology on Ff4(X )

553

31. Convolutions involving distributions

557

31.1. Tensor Product of Distributions

557

31.2. Elliptic Regularity

565

31.3. Appendix: Old Proof of Theorem 31.4

567

32. Pseudo-Dierential Operators on Euclidean space

571

32.1. Symbols and their operators

572

32.2. A more general symbol class

574

32.3. Schwartz Kernel Approach

583

32.4. Pseudo Dierential Operators

588

33. Elliptic pseudo dierential operators on Rg

600

34. Pseudo dierential operators on Compact Manifolds

604

35. Sobolev Spaces on P

608

35.1. Alternate Definition of Kn for n-integer

612

35.2. Scaled Spaces

614

35.3. General Properties of "Scaled space"

615

36. Compact and Fredholm Operators and the Spectral Theorem

618

36.1. Compact Operators

618

36.2. Hilbert Schmidt Operators

620

36.3. The Spectral Theorem for Self Adjoint Compact Operators

623

36.4. Structure of Compact Operators

627

36.5. Fredholm Operators

628

36.6. Tensor Product Spaces

633

37. Unbounded operators and quadratic forms

639

37.1. Unbounded operator basics

639

37.2. Lax-Milgram Methods

640

37.3. Close, symmetric, semi-bounded quadratic forms and self-adjoint

operators

642

37.4. Construction of positive self-adjoint operators

646

37.5. Applications to partial dierential equations

647

38. More Complex Variables: The Index

649

38.1. Unique Lifting Theorem

650

38.2. Path Lifting Property

650

39. Residue Theorem

657

39.1. Residue Theorem

658

39.2. Open Mapping Theorem

660

39.3. Applications of Residue Theorem

661

39.4. Isolated Singularity Theory

663

40. Conformal Equivalence

664

41. Find All Conformal Homeomorphisms of Y $ X

666

41.1. "Sketch of Proof of Riemann Mapping" Theorem

667

42. Radon Measures and F0([)

675

42.1. More Regularity Results

677

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BRUCE K. DRIVER

42.2. The Riesz Representation Theorem

680

42.3. The dual of F0([)

683

42.4. Special case of Riesz Theorem on [0> 1]

687

42.5. Applications

688

42.6. The General Riesz Representation by Daniell Integrals

690

42.7. Regularity Results

692

43. The Flow of a Vector Fields on Manifolds

698

Appendix A. Multinomial Theorems and Calculus Results

701

A.1. Multinomial Theorems and Product Rules

701

A.2. Taylor's Theorem

702

Appendix B. Zorn's Lemma and the Hausdor Maximal Principle

706

Appendix C. Cartheodory Method of Constructing Measures

710

C.1. Outer Measures

710

C.2. Carath?odory's Construction Theorem

712

C.3. Regularity results revisited

715

C.4. Construction of measures on a simple product space.

717

Appendix D. Nets

719

Appendix E. Infinite Dimensional Gaussian Measures

722

E.1. Finite Dimensional Examples and Results

723

E.2. Basic Infinite Dimensional Results

726

E.3. Guassian Measure for c2

730

E.4. Classical Wiener Measure

734

E.5. Basic Properties Wiener Measure

738

E.6. The Cameron-Martin Space and Theorem

740

E.7. Cameron-Martin Theorem

742

E.8. Exercises

746

Appendix F. Solutions to Selected Exercises

747

F.1. Section 2 Solutions

747

F.2. Section 3 Solutions

748

F.3. Section 5 Solutions

754

F.4. Section 6 Solutions

755

F.5. Section 7 Solutions

758

F.6. Section 8 Solutions

764

F.7. Section 9 Solutions

770

F.8. Section 10 Solutions

777

F.9. Section 11 Solutions

780

F.10. Section 12 Solutions

783

F.11. Section 13 Solutions

788

F.12. Section 14 Solutions

791

F.13. Section 15 Solutions

791

F.14. Section 16 Solutions

799

F.15. Section 17 Solutions

802

F.16. Section 18 Solutions

804

F.17. Size of c2 -- spaces.

820

F.18. Bochner Integral Problems form chapter 5 of first edition.

821

F.19. Section 19 Solutions

823

F.20. Section 20 Solutions

825

F.21. Section 21 Solutions

829

ANALYSIS TOOLS W ITH APPLICATIONS

ix

F.22. Section 24 Solutions

830

F.23. Section 26 Solutions

831

F.24. `Section 42 Solutions

833

F.25. Problems from Folland Sec. 7

833

F.26. Folland Chapter 2 problems

836

F.27. Folland Chapter 4 problems

836

Appendix G. Old Stu

844

G.1. Section 2

844

G.2. Section 3

847

G.3. Compactness on metric spaces

847

G.4. Compact Sets in Rq

849

G.5. Section 4

851

G.6. Section 5

852

G.7. Section 6:

854

G.8. Section 8

855

G.9. Section 9

866

G.10. Section 10

866

G.11. Section 11

870

G.12. Section 12

871

G.13. Section 13

876

G.14. Section 14

881

G.15. Section 15 old Stu

883

G.16. Signed measures

883

G.17. The Total Variation on an Algebra by B.

887

G.18. The Total Variation an Algebra by Z.

888

G.19. Old parts of Section 16

890

G.20. Old Absolute Continuity

890

G.21. Appendix: Absolute Continuity on an algebra by Z. (Delete?)

890

G.22. Other Hahn Decomposition Proofs

890

G.23. Old Dual to Os -- spaces

892

G.24. Section G.15

894

G.25. Section 16.4

894

G.26. Section 17

896

G.27. Old Urysohn's metrization Theorem

899

G.28. Section 18

901

G.29. Section 19

905

G.30. Section 20

907

G.31. Old Section 21

908

G.32. Old Section 27

909

G.33. Old Section 37

910

G.34. Old Section A

914

G.35. Old Section E.4

915

Appendix H. Record of Problems Graded

916

H.1. 240A F01

916

H.2. 240B W02

916

References

916

ANALYSIS TOOLS W ITH APPLICATIONS

1

1. Introduction

Not written as of yet. Topics to mention.

(1) A better and more general integral. (a) Convergence Theorems (b) Integration over diverse collection of sets. (See probability theory.) (c) Integration relative to dierent weights or densities including singular weights. (d) Characterization of dual spaces. (e) Completeness.

(2) Infinite dimensional Linear algebra. (3) ODE and PDE. (4) Harmonic and Fourier Analysis. (5) Probability Theory

2. Limits, sums, and other basics

2.1. Set Operations. Suppose that [ is a set. Let P([) or 2[ denote the power set of [> that is elements of P([) = 2[ are subsets of D= For D 5 2[ let

Df = [ \ D = {{ 5 [ : { 5@ D}

and more generally if D> E [ let

E \ D = {{ 5 E : { 5@ D}=

We also define the symmetric dierence of D and E by

D4E = (E \ D) ^ (D \ E) =

As usual if {D}5L is an indexed collection of subsets of [ we define the union and the intersection of this collection by

^5L D := {{ 5 [ : < 5 L 3 { 5 D} and _5L D := {{ 5 [ : { 5 D ; 5 L }=

Notation

2.1.

We

will

also

write

`

5L

D

for

^5L D

in

the

case

that

{D}5L

are pairwise disjoint, i.e. D _ D = > if 6= =

Notice that ^ is closely related to < and _ is closely related to ;= For example let {Dq}4 q=1 be a sequence of subsets from [ and define

{Dq i.o.} := {{ 5 [ : # {q : { 5 Dq} = 4} and {Dq a.a.} := {{ 5 [ : { 5 Dq for all q su!ciently large}.

(One should read {Dq i.o.} as Dq infinitely often and {Dq a.a.} as Dq almost always.) Then { 5 {Dq i.o.} i ;Q 5 N { 5 Dq which may be written as

{Dq a.a.} = ^4 Q=1 _qQ Dq=

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