MATH3011/4201 Analysis III/IV
This module gives a rigorous development and applications of a cornerstone of analysis, namely integration as developed though measure theory. The methods and results are fundamental to many areas of pure and applied mathematics.
The course starts with elements of the theory of Lebesgue measure and integration, with the aim of providing a rigorous basis for defining the most important examples of complete normed and inner product vector spaces, namely the Lebesgue spaces of integrable functions on the real line.
It proceeds with the study of elementary properties of Banach and Hilbert space.
Thirdly, it introduces Fourier series and its convergence theorems. These occur as a special case of the spectral theory of Sturm-Liouville problems, a boundary value problem for an ordinary differential equation of second order that has wide applications in physics and engineering.
Outline of Course
- Set theory: countable and uncountable sets, axiom of choice, the real numbers.
- Metric spaces: completeness, separability, compactness.
- Continuous functions: equicontinuity and Ascoli's theorem, the Stone-Weierstrass theorem.
- Measure: outer measure, measurability and sigma-algebras, Borel sets, Lebesgue measure.
- Measurable functions: properties of measurable functions, approximation by simple functions and by continuous functions.
- Integration: construction and properties of the integral, convergence theorems.
- Special properties of functions on the real line: absolute continuity, differentiation and integration.
- Elementary properties of Banach and Hilbert spaces: the Lebesgue spaces Lp, the space of continuous functions C(X), completeness and the Riesz-Fischer theorem, orthonormal bases of L².
- Basic harmonic analysis: Basic properties of Fourier series, convergence in L².