Mathematical Analysis Volume I
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1 The Zakon Series on Mathematical Analysis Mathematical Analysis Volume I Elias Zakon University of Windsor The Trillia Group West Lafayette, IN
2 Printed from e-media with permission by: Curran Associates, Inc. 57 Morehouse Lane Red Hook, NY Some format issues inherent in the e-media version may also appear in this print version. Anyone purchasing this book is granted a single-user license to the associated e-book; see for details. Mathematical Analysis I c 1975 Elias Zakon c 2004 Bradley J. Lucier and Tamara Zakon All rights reserved. Printed by Curran Associates, Inc. (2010) ISBN Published by the Trillia Group, West Lafayette, Indiana, USA. First published: May 20, This version released: July 11, Technical Typist: Betty Gick. Copy Editor: John Spiegelman. Logo: Miriam Bogdanic. The phrase The Trillia Group and the Trillia Group logo are trademarks of the Trillia Group. This book was prepared by Bradley J. Lucier and Tamara Zakon from a manuscript written by Elias Zakon. We intend to correct and update this work as needed. If you notice any mistakes in this work, please send to lucier@math.purdue.edu and they will be corrected in a later version. For any additional requests please contact the Trillia Group at info@trillia.com.
3 Contents Preface About the Author ix xi Chapter 1. Set Theory Sets and Operations on Sets. Quantifiers...1 Problems in Set Theory Relations. Mappings...8 Problems on Relations and Mappings Sequences Some Theorems on Countable Sets...18 Problems on Countable and Uncountable Sets Chapter 2. Real Numbers. Fields Axioms and Basic Definitions Natural Numbers. Induction...27 Problems on Natural Numbers and Induction Integers and Rationals Upper and Lower Bounds. Completeness...36 Problems on Upper and Lower Bounds Some Consequences of the Completeness Axiom Powers With Arbitrary Real Exponents. Irrationals...46 Problems on Roots, Powers, and Irrationals The Infinities. Upper and Lower Limits of Sequences...53 Problems on Upper and Lower Limits of Sequences in E...60 Chapter 3. Vector Spaces. Metric Spaces The Euclidean n-space, E n...63 Problems on Vectors in E n Lines and Planes in E n...71 Problems on Lines and Planes in E n...75 Starred sections may be omitted by beginners.
4 vi Contents 7. Intervals in E n...76 Problems on Intervals in E n Complex Numbers...80 Problems on Complex Numbers Vector Spaces. The Space C n. Euclidean Spaces...85 Problems on Linear Spaces Normed Linear Spaces...90 Problems on Normed Linear Spaces Metric Spaces Problems on Metric Spaces Open and Closed Sets. Neighborhoods Problems on Neighborhoods, Open and Closed Sets Bounded Sets. Diameters Problems on Boundedness and Diameters Cluster Points. Convergent Sequences Problems on Cluster Points and Convergence Operations on Convergent Sequences Problems on Limits of Sequences More on Cluster Points and Closed Sets. Density Problems on Cluster Points, Closed Sets, and Density Cauchy Sequences. Completeness Problems on Cauchy Sequences Chapter 4. Function Limits and Continuity Basic Definitions Problems on Limits and Continuity Some General Theorems on Limits and Continuity More Problems on Limits and Continuity Operations on Limits. Rational Functions Problems on Continuity of Vector-Valued Functions Infinite Limits. Operations in E Problems on Limits and Operations in E Monotone Functions Problems on Monotone Functions Compact Sets Problems on Compact Sets More on Compactness...192
5 Contents vii 8. Continuity on Compact Sets. Uniform Continuity Problems on Uniform Continuity; Continuity on Compact Sets The Intermediate Value Property Problems on the Darboux Property and Related Topics Arcs and Curves. Connected Sets Problems on Arcs, Curves, and Connected Sets Product Spaces. Double and Iterated Limits Problems on Double Limits and Product Spaces Sequences and Series of Functions Problems on Sequences and Series of Functions Absolutely Convergent Series. Power Series More Problems on Series of Functions Chapter 5. Differentiation and Antidifferentiation Derivatives of Functions of One Real Variable Problems on Derived Functions in One Variable Derivatives of Extended-Real Functions Problems on Derivatives of Extended-Real Functions L Hôpital s Rule Problems on L Hôpital s Rule Complex and Vector-Valued Functions on E Problems on Complex and Vector-Valued Functions on E Antiderivatives (Primitives, Integrals) Problems on Antiderivatives Differentials. Taylor s Theorem and Taylor s Series Problems on Taylor s Theorem The Total Variation (Length) of a Function f: E 1 E Problems on Total Variation and Graph Length Rectifiable Arcs. Absolute Continuity Problems on Absolute Continuity and Rectifiable Arcs Convergence Theorems in Differentiation and Integration Problems on Convergence in Differentiation and Integration Sufficient Condition of Integrability. Regulated Functions Problems on Regulated Functions Integral Definitions of Some Functions Problems on Exponential and Trigonometric Functions Index 341
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