Analysis II

As with the first, the second volume contains substantially more material than can be covered in a one-semester course. Such courses may omit many beautiful and well-grounded applications which connect broadly to many areas of mathematics. We of course hope that students will pursue this material independently; teachers may find it useful for undergraduate seminars. For an overview of the material presented, consult the table of contents and the chapter introductions. As before, we stress that doing the numerous exercises is indispensable for understanding the subject matter, and they also round out and amplify the main text. In writing this volume, we are indebted to the help of many. We especially thank our friends and colleagues Pavol Quittner and Gieri Simonett. They have not only meticulously reviewed the entire manuscript and assisted in weeding out errors but also, through their valuable suggestions for improvement, contributed essentially to the final version. We also extend great thanks to our staff for their careful perusal of the entire manuscript and for tracking errata and inaccuracies. Our most heartfelt thank extends again to our “typesetting perfectionist”, 1 without whose tireless effort this book would not look nearly so nice. We also thank Andreas for helping resolve hardware and software problems. Finally, we extend thanks to Thomas Hintermann and to Birkhauser for the good working relationship and their understanding of our desired deadlines. Sachverzeichnis......Page 0 Foreword......Page 5 Foreword to the English translation......Page 6 Chapter VI: Integral calculus in one variable......Page 7 Chapter VII: Multivariable differential calculus......Page 9 Chapter VIII: Line integrals......Page 11 Index......Page 12 Chapter VI: Integral calculus in one variable......Page 13 Staircase and jump continuous functions......Page 16 A characterization of jump continuous functions......Page 18 The Banach space of jump continuous functions......Page 19 Exercises......Page 20 The extension of uniformly continuous functions......Page 22 Bounded linear operators......Page 24 The continuous extension of bounded linear operators......Page 27 Exercises......Page 28 The integral of staircase functions......Page 29 The integral of jump continuous functions......Page 31 Riemann sums......Page 32 Exercises......Page 35 Integration of sequences of functions......Page 37 The oriented integral......Page 38 Positivity and monotony of integrals......Page 39 The first fundamental theorem of calculus......Page 42 The indefinite integral......Page 44 The mean value theorem for integrals......Page 45 Exercises......Page 46 Variable substitution......Page 50 Integration by parts......Page 52 The integrals of rational functions......Page 55 Exercises......Page 60 The Bernoulli numbers......Page 62 Recursion formulas......Page 64 The Bernoulli polynomials......Page 65 The Euler–Maclaurin sum formula......Page 66 Power sums......Page 68 Asymptotic equivalence......Page 69 The Riemann ζ function......Page 71 The trapezoid rule......Page 76 Exercises......Page 77 The L2 scalar product......Page 79 Approximating in the quadratic mean......Page 81 Orthonormal systems......Page 83 Integrating periodic functions......Page 84 Fourier coefficients......Page 85 Classical Fourier series......Page 86 Bessel’s inequality......Page 89 Complete orthonormal systems......Page 91 Piecewise continuously differentiable functions......Page 94 Exercises......Page 99 Improper integrals......Page 102 The integral comparison test for series......Page 105 Absolutely convergent integrals......Page 106 The majorant criterion......Page 107 Exercises......Page 109 Euler’s integral representation......Page 110 The gamma function on C\(-N)......Page 111 Gauss’s representation formula......Page 112 The reflection formula......Page 116 The logarithmic convexity of the gamma function......Page 117 Stirling’s formula......Page 120 The Euler beta integral......Page 122 Exercises......Page 124 Chapter VII: Multivariable differential calculus......Page 126 The completeness of L(E,F)......Page 129 Finite-dimensional Banach spaces......Page 130 Matrix representations......Page 133 The exponential map......Page 136 Linear differential equations......Page 139 Gronwall’s lemma......Page 140 The variation of constants formula......Page 142 Determinants and eigenvalues......Page 144 Fundamental matrices......Page 147 Second order linear differential equations......Page 151 Exercises......Page 156 The definition......Page 160 The derivative......Page 161 Directional derivatives......Page 163 Partial derivatives......Page 164 The Jacobi matrix......Page 166 A differentiability criterion......Page 167 The Riesz representation theorem......Page 169 The gradient......Page 170 Complex differentiability......Page 173 Exercises......Page 175 The chain rule......Page 177 The mean value theorem......Page 180 Necessary condition for local extrema......Page 182 Exercises......Page 183 Continuous multilinear maps......Page 184 The canonical isomorphism......Page 186 Symmetric multilinear maps......Page 187 The derivative of multilinear maps......Page 188 Exercises......Page 190 Definitions......Page 191 Higher order partial derivatives......Page 194 Taylor’s formula......Page 196 Functions of m variables......Page 197 Sufficient criterion for local extrema......Page 199 Exercises......Page 202 The continuity of Nemytskii operators......Page 206 The differentiability of Nemytskii operators......Page 208 The differentiability of parameter-dependent integrals......Page 211 Variational problems......Page 213 The Euler–Lagrange equation......Page 215 Classical mechanics......Page 218 Exercises......Page 220 The derivative of the inverse of linear maps......Page 223 The inverse function theorem......Page 225 Diffeomorphisms......Page 228 The solvability of nonlinear systems of equations......Page 229 Exercises......Page 230 Differentiable maps on product spaces......Page 232 The implicit function theorem......Page 234 Ordinary differential equations......Page 237 Separation of variables......Page 240 Lipschitz continuity and uniqueness......Page 244 The Picard–Lindelöf theorem......Page 246 Exercises......Page 251 Submanifolds of Rn......Page 253 The regular value theorem......Page 254 The immersion theorem......Page 255 Embeddings......Page 258 Local charts and parametrizations......Page 263 Change of charts......Page 266 Exercises......Page 267 The tangential in Rn......Page 271 The tangential space......Page 272 Characterization of the tangential space......Page 276 Differentiable maps......Page 277 The differential and the gradient......Page 280 Normals......Page 282 Constrained extrema......Page 283 Applications of Lagrange multipliers......Page 284 Exercises......Page 288 Chapter VIII: Line integrals......Page 290 The total variation......Page 292 Rectifiable paths......Page 293 Differentiable curves......Page 295 Rectifiable curves......Page 297 Exercises......Page 300 Unit tangent vectors......Page 303 Parametrization by arc length......Page 304 Oriented bases......Page 305 The Frenet n-frame......Page 306 Curvature of plane curves......Page 309 Instantaneous circles along curves......Page 311 The vector product......Page 313 The curvature and torsion of space curves......Page 314 Exercises......Page 315 Vector fields and Pfaff forms......Page 319 The canonical basis......Page 321 Exact forms and gradient fields......Page 323 The Poincaré lemma......Page 325 Dual operators......Page 327 Transformation rules......Page 328 Modules......Page 332 Exercises......Page 334 The definition......Page 337 Elementary properties......Page 339 The fundamental theorem of line integrals......Page 341 Simply connected sets......Page 343 The homotopy invariance of line integrals......Page 344 Exercises......Page 347 Complex line integrals......Page 350 Holomorphism......Page 353 The Cauchy integral theorem......Page 354 The orientation of circles......Page 355 The Cauchy integral formula......Page 356 Analytic functions......Page 357 Liouville’s theorem......Page 359 The Fresnel integral......Page 360 The maximum principle......Page 361 Harmonic functions......Page 362 Goursat’s theorem......Page 364 Exercises......Page 367 The Laurent expansion......Page 371 Removable singularities......Page 375 Isolated singularities......Page 376 Simple poles......Page 379 The winding number......Page 381 The continuity of the winding number......Page 385 The generalized Cauchy integral theorem......Page 387 The residue theorem......Page 389 Fourier integrals......Page 390 Exercises......Page 394 References......Page 397 Index......Page 399 This book is the first of a three volume introduction to analysis. It is distinguished by its modern and clear presentation, concentrating always on the essential concepts. In contrast to most other textbooks, there is no artificial separation between the theories of one variable and that of many variables. Emphasis is placed on the early development of a solid foundation in topology. As well, the basics of complex analysis are covered. This book is directed primarily to the students and instructors of beginning courses in analysis. But, with the many examples, exercises and the supplementary material, it is also suitable for self-study, as preparation for advanced study, and as a basis for other research in mathematics and physics."This textbook provides an outstanding introduction to analysis. It is distinguished by its high level of presentation and its focus on the essential.''Zeitschrift für Analysis und ihre Anwendung 18, No. 4 (G. Berger, review of the first German edition)"One advantage of this presentation is that the power of the abstract concepts are convincingly demonstrated using concrete applications.''W. Grölz, review of the first German edition