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Core Module in Analysis

Dr Neill Robertson
1st semester 2019
20 credits / 30 lectures

The central theme in Mathematical Analysis is the study of limiting processes. There are many problems in Mathematics that cannot be solved explicitly, but where it is relatively easy to find a sequence of approximate solutions. An obvious strategy is to look for a limit (in some sense) of these approximate solutions. The purpose of this course is to study some of the basic tools that are used to do this. It builds on elements of advanced real analysis and metric spaces and introduces some of the basic techniques of measure theory and functional analysis. Outline:

  • Revision of Riemann integration
  • Inner and outer measures on the real line
  • Lebesgue measure on R
  • The existence of non-measurable sets
  • Algebras of sets, sigma algebras
  • Measurable functions
  • Measures and measure spaces
  • Completeness of measures, Borel measures
  • Integrable functions, integrals
  • Fatou’s lemma, the monotone and dominated convergence theorems
  • Signed measures, the Radon-Nikodym theorem
  • Norms and seminorms, Banach spaces
  • The Holder and Minkowski inequalities, Lp-spaces
  • Linear maps and dual spaces, the Riesz representation theorem
  • Modes of convergence

Prerequisites:

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