## Analysis ILogical thinking, the analysis of complex relationships, the recognition of und- lying simple structures which are common to a multitude of problems — these are the skills which are needed to do mathematics, and their development is the main goal of mathematics education. Of course, these skills cannot be learned ‘in a vacuum’. Only a continuous struggle with concrete problems and a striving for deep understanding leads to success. A good measure of abstraction is needed to allow one to concentrate on the essential, without being distracted by appearances and irrelevancies. The present book strives for clarity and transparency. Right from the beg- ning, it requires from the reader a willingness to deal with abstract concepts, as well as a considerable measure of self-initiative. For these e?orts, the reader will be richly rewarded in his or her mathematical thinking abilities, and will possess the foundation needed for a deeper penetration into mathematics and its applications. Thisbookisthe?rstvolumeofathreevolumeintroductiontoanalysis.It- veloped from courses that the authors have taught over the last twenty six years at theUniversitiesofBochum,Kiel,Zurich,BaselandKassel.Sincewehopethatthis book will be used also for self-study and supplementary reading, we have included far more material than can be covered in a three semester sequence. This allows us to provide a wide overview of the subject and to present the many beautiful and important applications of the theory. We also demonstrate that mathematics possesses, not only elegance and inner beauty, but also provides e?cient methods for the solution of concrete problems. |

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### Contents

17 | |

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XVI | 22 |

XVII | 25 |

XVIII | 28 |

XIX | 30 |

XX | 33 |

XXI | 34 |

XXII | 38 |

XXIII | 45 |

XXIV | 46 |

XXV | 47 |

XXVI | 48 |

XXVII | 51 |

XXIX | 53 |

XXX | 54 |

XXXI | 55 |

XXXII | 57 |

XXXIII | 61 |

XXXIV | 64 |

XXXV | 66 |

XXXVI | 68 |

XXXVII | 70 |

XXXVIII | 72 |

XXXIX | 75 |

XL | 76 |

XLI | 77 |

XLII | 83 |

XLIII | 84 |

XLIV | 87 |

XLVI | 90 |

XLVII | 91 |

XLVIII | 93 |

XLIX | 94 |

L | 95 |

LI | 96 |

LII | 98 |

LIII | 99 |

LIV | 102 |

LV | 103 |

LVI | 105 |

LVII | 107 |

LVIII | 110 |

LX | 111 |

LXI | 114 |

LXII | 116 |

LXIII | 118 |

LXIV | 119 |

LXV | 121 |

LXVI | 122 |

LXVII | 123 |

LXVIII | 128 |

LXIX | 130 |

LXX | 131 |

LXXI | 133 |

LXXII | 134 |

LXXIII | 136 |

LXXIV | 137 |

LXXV | 140 |

LXXVI | 142 |

LXXVII | 143 |

LXXVIII | 147 |

LXXIX | 148 |

LXXX | 149 |

LXXXI | 150 |

LXXXII | 152 |

LXXXIII | 153 |

LXXXIV | 155 |

LXXXV | 156 |

LXXXVI | 158 |

LXXXVII | 162 |

LXXXVIII | 163 |

LXXXIX | 168 |

XCI | 169 |

XCII | 171 |

XCIII | 174 |

XCIV | 175 |

XCV | 176 |

XCVI | 182 |

XCVII | 183 |

XCVIII | 184 |

XCIX | 185 |

C | 186 |

CI | 191 |

CII | 194 |

CIII | 195 |

CXIV | 218 |

CXV | 223 |

CXVII | 227 |

CXVIII | 231 |

CXX | 232 |

CXXI | 234 |

CXXII | 235 |

CXXIII | 236 |

CXXIV | 237 |

CXXV | 238 |

CXXVI | 240 |

CXXVII | 243 |

CXXVIII | 244 |

CXXIX | 249 |

CXXX | 250 |

CXXXI | 251 |

CXXXIII | 252 |

CXXXIV | 255 |

CXXXV | 257 |

CXXXVI | 258 |

CXXXVII | 262 |

CXXXVIII | 263 |

CXXXIX | 264 |

CXLI | 267 |

CXLII | 270 |

CXLIII | 271 |

CXLIV | 273 |

CXLV | 276 |

CXLVII | 279 |

CXLVIII | 280 |

CXLIX | 282 |

CL | 284 |

CLI | 288 |

CLII | 289 |

CLIII | 290 |

CLIV | 292 |

CLV | 293 |

CLVI | 294 |

CLVII | 298 |

CLIX | 300 |

CLX | 301 |

CLXI | 303 |

CLXII | 304 |

CLXIII | 305 |

CLXIV | 306 |

CLXV | 312 |

CLXVI | 316 |

CLXVII | 317 |

CLXVIII | 318 |

CLXIX | 321 |

CLXX | 324 |

CLXXI | 327 |

CLXXII | 328 |

CLXXIII | 329 |

CLXXIV | 334 |

CLXXV | 335 |

CLXXVI | 337 |

CLXXVII | 339 |

CLXXVIII | 343 |

CLXXIX | 344 |

CLXXX | 349 |

CLXXXI | 350 |

CLXXXII | 354 |

CLXXXIII | 359 |

CLXXXIV | 362 |

CLXXXV | 363 |

CLXXXVI | 365 |

CLXXXVII | 366 |

CLXXXVIII | 369 |

CXCI | 371 |

CXCII | 372 |

CXCIII | 376 |

CXCIV | 377 |

CXCV | 379 |

CXCVI | 380 |

CXCVII | 381 |

CXCVIII | 385 |

CXCIX | 389 |

CC | 390 |

CCI | 392 |

CCII | 395 |

CCIII | 397 |

CCIV | 400 |

CCV | 403 |

CCVII | 409 |

410 | |

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### Common terms and phrases

absolutely convergent algebra analytic axioms Banach space bijective bounded called Cauchy sequence claim follows closed cluster point commutative compact complete consider continuous functions convergent sequence converges uniformly convex Corollary countable defined differentiable equation equivalent Example Exercise exponential function f is continuous finite follows directly follows from Proposition function f Hence Hint homomorphism implies induction infinite injective inner product space interval isomorphism Let f limit point linear Lipschitz continuous mean value theorem metric space natural numbers neighborhood nonempty subset normed vector space null sequence operation pointwise polynomial power series Proof Let Proof This follows Proposition Let Prove the following radius of convergence rational numbers real numbers Remark series XL statement subspace sup(A surjective topological space triangle inequality true uniform convergence unique zero