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. 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. With this book, well-known author Thomas Korner 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 technique.
The real line A first philosophical interlude Other versions of the fundamental axiom Higher dimensions Sums and suchlike $\heartsuit$ Differentiation Local Taylor theorems The Riemann integral Developments and limitations of the Riemann integral $\heartsuit$ Metric spaces Complete metric spaces Contraction mappings and differential equations Inverse and implicit functions Completion Appendices Executive summary Exercises Bibliography Index.