Mathematical Analysis Fundamentals

The author’s goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options.

Deeper discussion of the basic concept of convergence for the system of real numbers, pointing out its specific features, and for metric spaces

Presentation of Riemann integration and its place in the whole integration theory for single variable, including the Kurzweil-Henstock integration

Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus.

1. Sets and Proofs 2. Numbers 3. Convergence 4. Point Set Topology 5. Continuity 6. The Space 7. Differentiation 8. Bounded Variation 9. Reimann Integration 10. Generalizations of Reimann Integration 11. Transcendental Functions 12. Fourier Series and Integers