Mathematical Analysis - Period 4+5

This course treats the rigorous mathematical theory behind Calculus: limits, continuity, linear approximation, differentiability, integrability, and the mutual relation between these concepts. The mathematical tools that are necessary for formulating and proving the essential results of Calculus are first presented in the context of real valued sequences and real valued functions of a real variable, in such a way that everything can later be generalised (to Y-valued functions of variables in X, with X and Y Banach spaces). The space C[a,b] of real valued continuous functions on an interval [a,b] will appear as the first example of such a Banach space.

Starting point of the course are an ancient iterative scheme for solving equations, and the fundamental properties of (the set of) real numbers. Highlights: a fairly complete exposition of power series directly based on a systematic algebraic approach for monomials, and an early introduction of the Implicit Function Theorem via a contraction argument and the Banach Fixed Point Theorem.

Topics covered:
1. Cauchy sequences, convergence, limits;
2. Completeness of the real numbers; theorem of Bolzano-Weierstrass;
3. Continuity and uniform continuity;
4. The concept of differentiability (including differentiability of power series);
5. The concept of Riemann integrability (including Riemann integrability of monotone and uniformly continuous functions);
6. The language of metric topology;
7. Completeness of the space C[a,b]; uniform convergence;
8. The Banach Fixed Point Theorem (with applications to integral and differential equations, and the implicit function theorem).

Contact hours listed under a course description may vary due to the combination of lecture-based and independent work required for each course therefore, CEA's recommended credits are based on the ECTS credits assigned by VU Amsterdam. 1 ECTS equals 28 contact hours assigned by VU Amsterdam.