Introduction to Analysis — ANT

Instructor: 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  Rⁿ (Implicit and Inverse function theorems)

4. Integration

• Riemann-Stieltjes integrals (integration by parts, fundamental theorem of integral calculus)
• Integration in Rⁿ (Jordan measure)

5. Fourier series

• Series of functions (uniform, pointwise and L₂ convergence)
• Arzela-Ascoli theorem, compact sets in  C(X)
• Fourier series (Fourier coefficients, Bessel's inequality, Parseval formula, applications)
• Theorem of Fejér and Riesz-Fisher
• Weierstrass approximation theorem