Korner T.W. A Companion to Analysis: A Second First and First Second Course in Analysis

Cambridge, 1991. 598 p
Many students acquire knowledge of a large number of theorems and methods of calculus without being able to say how they work together. This book provides those students with the coherent account that they need. A Companion to Analysis explains the problems that must be resolved in order to procure a rigorous development of the calculus and shows the student how to deal with those problems. P Starting with the real line, the book moves on to finite-dimensional spaces and then to metric spaces. Readers who work through this text will be ready for courses such as measure theory, functional analysis, complex analysis, and differential geometry. Moreover, they will be well on the road that leads from mathematics student to mathematician. P With this book, well-known author Thomas Körner provides able and hard-working students a great text for independent study or for an advanced undergraduate or first-level graduate course. It includes many stimulating exercises. An appendix contains a large number of accessible but non-routine problems that will help students advance their knowledge and improve their techniqueThe Real Line
Why do we bother?
Limits
Continuity
The fundamental axiom
The axiom of Archimedes
Lion hunting
The mean value inequality
Full circle
Are the real numbers unique?
A First Philosophical Interlude
Is the intermediate value theorem obvious?
Other Versions of the Fundamental Axiom
The supremum
The Bolzano-Weierstrass theorem
Some general remarks
Higher Dimensions Bolzano-Weierstrass in higher dimensions
Open and closed sets
A central theorem of analysis
The mean value theorem
Uniform continuity
The general principle of convergence
Sums and Suchlike
Comparison tests
Conditional convergence
Interchanging limits
The exponential function
The trigonometric functions
The logarithm
Powers
The fundamental theorem of algebra
Differentiation
Preliminaries
The operator norm and the chain rule
The mean value inequality in higher dimensions
Local Taylor Theorems
Some one dimensional Taylor theorems
Some many dimensional local Taylor theorems
Critical points
The Riemann Integral
Where is the problem?
Riemann integration
Integrals of continuous functions
First steps in the calculus of variations
Vector-valued integrals
Developments and limitations of the Riemann integral
Why go further?
mproper integrals
ntegrals over areas
The Riemann-Stieltjes integral
How long is a piece of string?
Metric Spaces
Sphere packing
Shannon's theorem
Metric spaces
Norms and the interaction of algebra and analysis
Geodesies
Complete Metric Spaces
Completeness
The Bolzano-Weierstrass property
The uniform norm
Uniform convergence
Power series
Fourier series
Contraction Mappings and Differential Equations
Banach's contraction mapping theorem
Existence of solutions of differential equations
Local to global
Green's function solutions
Inverse and Implicit Functions
The inverse function theorem
The implicit function theorem
Lagrange multipliers
Completion What is the correct question?
The solution
Why do we construct the reals?
How do we construct the reals?
Paradise lost?
Appendix A. Ordered Fields
Appendix B. Countability
Appendix C. The care and treatment of counterexamples
Appendix D. A More General View of Limits
Appendix E. Traditional Partial Derivatives
Appendix F. Another approach to the inverse function theorem
Appendix G. Completing Ordered Fields
Appendix H. Constructive Analysis
Appendix I. Miscellany
Appendix J. Executive Summary
Appendix K. Exercises