Analysis I

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Springer Science & Business Media, Mar 14, 2006 - Mathematics - 426 pages
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Logical 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

II
xv
III
7
IV
8
VI
9
VII
11
VIII
14
IX
15
X
16
CIV
198
CVI
200
CVII
203
CVIII
209
CIX
210
CX
212
CXI
213
CXII
216

XII
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XIII
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XIV
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XV
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XVI
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XVII
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XVIII
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XIX
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XX
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XXI
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XXII
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XXIII
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XXIV
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XXVI
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XXVII
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XXIX
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XXX
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XXXI
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XXXII
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XXXIII
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XXXIV
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XXXV
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XXXVI
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XXXVII
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XXXVIII
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XXXIX
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XL
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XLI
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XLII
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XLIII
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XLIV
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XLVI
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XLVII
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XLVIII
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XLIX
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L
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LI
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LII
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LIII
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LIV
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LV
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LVI
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LVII
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LVIII
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LX
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LXI
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LXII
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LXIII
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LXIV
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LXV
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LXVI
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LXVII
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LXVIII
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LXIX
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LXX
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LXXI
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LXXII
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LXXIII
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LXXIV
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LXXV
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LXXVI
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LXXVII
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LXXVIII
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LXXIX
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LXXX
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LXXXI
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LXXXII
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LXXXIII
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LXXXIV
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LXXXV
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LXXXVI
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LXXXVII
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LXXXVIII
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LXXXIX
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XCI
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XCII
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XCIII
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XCIV
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XCV
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XCVI
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XCVII
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XCVIII
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XCIX
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C
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CI
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CII
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CIII
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CXIV
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CXV
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CXVII
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CXVIII
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CXX
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CXXI
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CXXII
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CXXIII
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CXXIV
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CXXV
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CXXVI
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CXXVII
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CXXVIII
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CXXIX
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CXXX
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CXXXI
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CXXXIII
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CXXXIV
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CXXXV
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CXXXVI
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CXXXVII
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CXXXVIII
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CXXXIX
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CXLI
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CXLII
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CXLIII
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CXLIV
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CXLV
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CXLVII
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CXLVIII
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CXLIX
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CL
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CLI
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CLII
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CLIII
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CLIV
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CLV
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CLVI
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CLVII
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CLIX
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CLX
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CLXI
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CLXII
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CLXIII
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CLXIV
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CLXV
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CLXVI
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CLXVII
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CLXVIII
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CLXIX
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CLXX
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CLXXI
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CLXXII
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CLXXIII
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CLXXIV
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CLXXV
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CLXXVI
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CLXXVII
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CLXXVIII
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CLXXIX
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CLXXX
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CLXXXI
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CLXXXII
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CLXXXIII
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CLXXXIV
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CLXXXV
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CLXXXVI
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CLXXXVII
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CLXXXVIII
369
CXCI
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CXCII
372
CXCIII
376
CXCIV
377
CXCV
379
CXCVI
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CXCVII
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CXCVIII
385
CXCIX
389
CC
390
CCI
392
CCII
395
CCIII
397
CCIV
400
CCV
403
CCVII
409
CCVIII
410
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