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Core Module in Analysis

Dr Neill Robertson
1st semester 2018
20 credits / 30 lectures

Many problems in Mathematics cannot be solved explicitly. The usual approach is to exhibit a sequence of approximate solutions and to study its convergence. The solution to the problem is then obtain as a limit (in some sense) of some subsequence of the sequence of approximate solutions. The central theme in Mathematical Analysis is the study of limiting processes and the purpose of this course is to study some fundamental tools which help to understand those limiting processes. This course is built upon elements of advanced real analysis and metric spaces. The course is divided essentially in two parts. The first part is on basic measure theory and integration including theorems like Beppo-Levi, Fatou’s lemma, the dominated convergence theorem and the Lp spaces. The second part of the course starts with the abstract Functional Analysis including the four pillar theorems which are the Hahn-Banach and its geometric forms, the Banach-Steinhaus, the open mapping and the closed graph theorems. This part involves also the notions of weak convergence in Banach spaces and their consequences together with some function spaces and their topological properties.

  1. Basic measure theory
  • Algebras of sets, sigma algebras, F-sigma and G-delta sets
  • Measures, measurable functions, Borel measures, the Lebesque measure
  • Integration, Fatou’s Lemma, Monotone and Dominated convergence theorems
  • Littlewood’s three principles: (i) every measurable set is nearly a union of intervals, (ii) every measurable function is nearly continuous, and (iii) every convergent sequence of measurable functions is nearly uniformly convergent.
  • Product measures, Tonelli and Fubini theorems
  1. Basic functional analysis
  • Lp spaces, Holder and Minkowski inequalities
  • Norms, seminorms, convergence and completeness
  • Linear operators, linear functionals
  • The Hahn-Banach and Banach-Steinhaus theorems, convexity
  • Open mapping and closed graph theorems
  1. Distributions and spaces of functions
  • Approximation of functions in Lp by compactly supported smooth functions (or differentiable functions or continuous functions)
  • The Schwarz class of functions
  • Tempered distributions as the dual space of the Schwarz class
  • The Dirac delta function
  1. Topics if time allows
  • Riesz representation theorem
  • Radon-Nikodym the​orem
  • theory of the Fourier transform

Prerequisites:

The 3rd year module 3MS (Metric Spaces), or equivalent, is required. The 3rd year module 3TN (Topics in Analysis) is recommended.