Analysis I Prelim topics

Topics

  • Solid grounding in undergraduate analysis at the level of Rudin’s analysis book.

  • Measures (outer measure on R, measurable spaces and functions, properties of measures, Lebesgue measure, convergence of measurable functions, Egorov's and Luzin's theorems)

  • Integration (integration with respect to a measure, monotone convergence theorem, limits of integrals and integrals of limits, dominated convergence theorem, Riemann and Lebesgue integrals)
  • Differentiation (Hardy-Littlewood maximal function and inequality, Lebesgue differentiation theorem)
  • Product measures (products of measure spaces, iterated integrals, Tonelli's and Fubini's theorems, Lebesgue measure/integration on R^n)
  • Banach spaces (normed vector spaces, bounded linear maps, linear functionals, Hahn-Banach theorems, consequences of Baire's theorem (open mapping theorem, inverse mapping theorem, closed graph theorem, and principle of uniform boundedness))

  • L^p spaces (Holder's and Minkowski's inequalities, L^p as a Banach space, duality).

Main References

  1. Walter Rudin, “Principles of Mathematical Analysis”
  2. S. Axler, ``Measure, integration & real analysis".
  3. H. Royden, P. Fitzpatrick, ``Real analysis" (4th edition).