Instructor: Dr. Károly BÖRÖCZKY

Text: handouts,

Prerequisite: Calculus

Course description: This course is actually an introduction which however, depending on the background of the class may quickly reach the most important specific areas of analysis.

Topics:

  Metric spaces, topology of metric spaces
  Sequences, convergence, Cauchy sequences, limsup and liminf
  Bolzano-Weierstrass theorem
  Complete metric spaces, Banach fixed point theorem
  Continuous functions, compact and connected sets
  Infinite series, convergent and absolutely convergent series
  Convergence tests, power series
  Cauchy-Hadamard formula
  Limit of functions, differentiation
  The Fundamental Theorem of Calculus
  Taylor series
  Differentiation of multivariable functions, partial derivatives
  Implicit and inverse function theorems
  Riemann-Stieltjes integrals, basic properties
  Integration in  Rⁿ  --- introduction
  Uniform, pointwise and $L_2$-convergence
  Arzela-Ascoli theorem, compact sets in $C(X)$
  Fourier series, various convergence theorems
  Theorem of Fejér and Riesz-Fisher
  Applications