Introduction to Analysis ANT
Instructor: Dr. András STIPSICZ and Dr. Szilárd SZABÓ
Text: handouts
Prerequisite: Calculus
Course description: After quickly reviewing basic notions and concepts
in analysis, we turn to the study of sequences of functions
and prove rudiments of Fourier series. Topological concepts are also
discussed in detail.
Topics:
1. Basic calculus

metrics and sequences

convergent and Cauchy sequences, complete metric spaces

Basic topology (open and closed sets, compact sets, connectedness)

Continuous functions
2. Infinite series

absolute and conditional convergence

Tests for convergence

Power series
3. Differentiation, multivariable calculus

Fundamental theorem of calculus

Differentiation in R^{n} (Implicit
and Inverse function theorems)
4. Integration

RiemannStieltjes integrals (integration by parts, fundamental theorem
of integral calculus)

Integration in R^{n }(Jordan measure)
5. Fourier series

Series of functions (uniform, pointwise and L_{2} convergence)

ArzelaAscoli theorem, compact sets in C(X)

Fourier series (Fourier coefficients, Bessel's inequality, Parseval formula,
applications)

Theorem of Fejér and RieszFisher

Weierstrass approximation theorem