<br><br><br><br> Real and functıonal analysıs<br><br>Thırd edıtıon <br><br>Serge lang<br><br><br>Contents<br>PART ONE<br>General Topology ............................................... 1<br>CHAPTER I<br>Sets ............................................................. 3<br>§1. Some Basic Terminology ........................................ 3<br>§2. Denumerable Sets .............................................. 7<br>§3. Zorn’s Lemma ................................................. 10<br>CHAPTER II<br>Topological Spaces ............................................... 17<br>§1. Open and Closed Sets .......................................... 17<br>§2. Connected Sets ................................................ 27<br>§3. Compact Spaces ............................................... 31<br>§4. Separation by Continuous Functions ............................. 40<br>§5. Exercises ...................................................... 43<br>CHAPTER III<br>Continuous Functions on Compact Sets ............................. 51<br>§1. The Stone-Weierstrass Theorem ................................. 51<br>§2. Ideals of Continuous Functions .................................. 55<br>§3. Ascoli’s Theorem .............................................. 57<br>§4. Exercises ...................................................... 59<br>PART TWO<br>Banach and Hilbert Spaces ..................................... 63<br>CHAPTER IV H<br>Banach Spaces .................................................. 65<br>§1. Definitions, the Dual Space, and the Hahn-Banach Theorem ....... 65<br>§2. Banach Algebras .............................................. 72<br>§3. The Linear Extension Theorem .................................. 75<br>§4. Completion of a Normed Vector Space ........................... 76<br>§5. Spaces with Operators ......................................... 81<br>Appendix: Convex Sets ......................................... 83<br>1. The Krein-Milman Theorem ................................. 83<br>2. Mazur’s Theorem ........................................... 88<br>§6. Exercises ..................................................... 91<br>CHAPTER V<br>Hilbert Space .................................................... 95<br>§1. Hermitian Forms .............................................. 95<br>§2. Functionals and Operators ..................................... 104<br>§3. Exercises ..................................................... 107<br>PART THREE<br>Integration ..................................................... 109<br>CHAPTER VI<br>The General Integral ............................................. Ill<br>§1. Measured Spaces, Measurable Maps, and Positive Measures ....... 112<br>§2. The Integral of Step Maps ..................................... 126<br>§3. The L:-Completion ........................................... 128<br>§4. Properties of the Integral: First Part ............................ 134<br>§5. Properties of the Integral: Second Part .......................... 137<br>§6. Approximations .............................................. 147<br>§7. Extension of Positive Measures from Algebras to cr-Algebras ....... 153<br>§8. Product Measures and Integration on a Product Space ..:......... 158<br>§9. The Lebesgue Integral in R" ................................... 166<br>§10. Exercises .................................................... 172<br>CHAPTER VII<br>Duality and Representation Theorems .............................. 181<br>§1. The Hilbert Space L2(n) ........................................ 181<br>§2. Duality Between Ll(n) and L°°(;u) ................................ 185<br>§3. Complex and Vectorial Measures ................................ 195<br>§4. Complex or Vectorial Measures and Duality ...................... 204<br>§5. The L" Spaces, 1 < p < oo ...................................... 209<br>§6. The Law of Large Numbers .................................... 213<br>§7. Exercises ..................................................... 217<br>CHAPTER VIII<br>Some Applications of Integration ................................... 223<br>§1. Convolution .................................................. 223<br>§2. Continuity and Differentiation Under the Integral Sign ............. 225<br>§3. Dirac Sequences ............................................... 227<br>§4. The Schwartz Space and Fourier Transform ...................... 236<br>§5. The Fourier Inversion Formula ................................. 241<br>§6. The Poisson Summation Formula ............................... 243<br>§7. An Example of Fourier Transform Not in the Schwartz Space ....... 244<br>§8. Exercises ..................................................... 247<br>CHAPTER IX<br>Integration and Measures on Locally Compact Spaces ............... 251<br>§1. Positive and Bounded Functionals on CC(X) ...................... 252<br>§2. Positive Functionals as Integrals ................................. 255<br>§3. Regular Positive Measures ...................................... 265<br>§4. Bounded Functionals as Integrals ................................ 267<br>§5. Localization of a Measure and of the Integral ..................... 269<br>§6. Product Measures on Locally Compact Spaces .................... 272<br>§7. Exercises ..................................................... 274<br>CHAPTER X<br>Riemann-Stieltjes Integral and Measure ............................ 278<br>§1. Functions of Bounded Variation and the Stieltjes Integral .......... 278<br>§2. Applications to Fourier Analysis ................................. 287<br>§3. Exercises ..................................................... 294<br>CHAPTER XI<br>Distributions ..................................................... 295<br>§1. Definition and Examples ....................................... 295<br>§2. Support and Localization ....................................... 299<br>§3. Derivation of Distributions ..................................... 303<br>§4. Distributions with Discrete Support .............................. " 304<br>CHAPTER XII<br>Integration on Locally Compact Groups ............................. 308<br>§1. Topological Groups ............................................ 308<br>§2. The Haar Integral, Uniqueness .................................. 313<br>§3. Existence of the Haar Integral ................................... 319<br>§4. Measures on Factor Groups and Homogeneous Spaces ............. 322<br>§5. Exercises ..................................................... 326<br>PART FOUR<br>Calculus ......................................................... 329<br>CHAPTER XIII<br>Differential Calculus .............................................. 331<br>§1. Integration in One Variable ................................... 331<br>§2. The Derivative as a Linear Map ............................... 333<br>§3. Properties of the Derivative ................................... 335<br>§4. Mean Value Theorem ........................................ 340<br>§5. The Second Derivative ........................................ 343<br>§6. Higher Derivatives and Taylor’s Formula ....................... 346<br>§7. Partial Derivatives ........................................... 351<br>§8. Differentiating Under the Integral Sign .......................... 355<br>§9. Differentiation of Sequences ................................... 356<br>§10. Exercises .................................................... 357<br>CHAPTER XIV<br>Inverse Mappings and Differential Equations ....................... 360<br>§1. The Inverse Mapping Theorem ................................. 360<br>§2. The Implicit Mapping Theorem ................................. 364<br>§3. Existence Theorem for Differential Equations ..................... 365<br>§4.

Local Dependence on Initial Conditions ......................... 371<br>§5. Global Smoothness of the Flow ................................. 376<br>§6. Exercises ..................................................... 379<br>PART FIVE<br>Functional Analysis ............................................ 385<br>CHAPTER XV<br>The Open Mapping Theorem, Factor Spaces, and Duality ............ 387<br>§1. The Open Mapping Theorem ................................... 387<br>§2. Orthogonality ................................................ 391<br>§3. Applications of the Open Mapping Theorem ..................... 395<br>CHAPTER XVI<br>The Spectrum .........................................;.-........ 400<br>§1. The Gelfand-Mazur Theorem .................................. 400<br>§2. The Gelfand Transform ........................................ 407<br>§3. C*-Algebras .................................................. 409<br>§4. Exercises ....................................................... 412<br>CHAPTER XVII<br>Compact and Fredholm Operators ................................. 415<br>§1. Compact Operators ........................................... 415<br>§2. Fredholm Operators and the Index .............................. 417<br>§3. Spectral Theorem for Compact Operators ........................ 426<br>§4. Application to Integral Equations ............................... 432<br>§5. Exercises ..................................................... 433<br>CHAPTER XVIII<br>Spectral Theorem for Bounded Hermitian Operators ................. 438<br>§1. Hermitian and Unitary Operators ............................... 438<br>§2. Positive Hermitian Operators ................................... 439<br>§3. The Spectral Theorem for Compact Hermitian Operators .......... 442<br>§4. The Spectral Theorem for Hermitian Operators ................... 444<br>§5. Orthogonal Projections ........................................ 449<br>§6. Schur’s Lemma ............................................... 452<br>§7. Polar Decomposition of Endomorphisms ......................... 453<br>§8. The Morse-Palais Lemma ..................................... 455<br>§9. Exercises ..................................................... 458<br>CHAPTER XIX<br>Further Spectral Theorems ....................................... 464<br>§1. Projection Functions of Operators .............................. 464<br>§2. Self-Adjoint Operators ......................................... 469<br>§3. Example: The Laplace Operator in the Plane ..................... 476<br>CHAPTER XX<br>Spectral Measures ............................................... 480<br>§1. Definition of the Spectral Measure .............................. 480<br>§2. Uniqueness of the Spectral Measure:<br>the Titchmarsh-Kodaira Formula ............................... 485<br>§3. Unbounded Functions of Operators ............................. 488<br>§4. Spectral Families of Projections ................................. 490<br>§5. The Spectral Integral as Stieltjes Integral ......................... 491<br>§6. Exercises ..................................................... 492<br>PART SIX<br>Global Analysis ................................................ 495<br>CHAPTER XXI<br>Local Integration of Differential Forms ............................. 497<br>§1. Sets of Measure 0 ............................................. -497<br>§2. Change of Variables Formula ................................... 498<br>§3. Differential Forms ............................................. 507<br>§4. Inverse Image of a Form ....................................... 512<br>§5. Appendix .................................................... 516<br>CHAPTER XXII<br>Manifolds ....................................................... 523<br>§1. Atlases, Charts, Morphisms ..................................... 523<br>§2. Submanifolds ................................................. 527<br>§3. Tangent Spaces ............................................... 533<br>§4. Partitions of Unity ............................................ 536<br>§5. Manifolds with Boundary ...................................... 539<br>§6. Vector Fields and Global Differential Equations .................. 543<br>CHAPTER XXIII<br>Integration and Measures on Manifolds ............................. 547<br>§1. Differential Forms on Manifolds ................................. 547<br>§2. Orientation ................................................... 551<br>§3. The Measure Associated with a Differential Form ................. 553<br>§4. Stokes’ Theorem for a Rectangular Simplex ....................... 555<br>§5. Stokes’ Theorem on a Manifold ................................. 558<br>§6. Stokes’ Theorem with Singularities ............................... 561<br>Bibliography .................................................... 569<br>Table of Notation ............................................... 572<br>Index ........................................................... 575<br>