Math 541: Real Analysis

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Catalog Information

Title

Real Analysis.

Credit Hours

3

Prerequisite

Math 315, 343; 214 or 316.

Description

Rigorous treatment of differentiation and integration theory; Lebesque measure; Banach spaces.

Desired Learning Outcomes

Math 541 is currently the first half of a two-semester sequence on Lebesgue integration in Euclidean space and several related topics, but it is proposed that it become a one-semester course specifically on Lebesgue integration in Euclidean space and Fourier transforms.

Prerequisites

Minimal learning outcomes

  1. Lebesgue measure on Rn
    • Inner and outer measures
    • Construction of Lebesgue measure
    • Properties of Lebesgue measure
      • Effect of basic set operations
      • Limiting properties
      • Its domain
      • Approximation properties
      • Sets of outer measure zero
      • Invariance w.r.t. isometries
      • Effect of dilations
    • Existence of nonmeasurable sets
  2. Lebesgue integration on Rn
    • Measurable functions
    • Simple functions
    • Approximation of measurable functions with simple functions
    • The extended reals
    • Integrating nonnegative functions
    • Integrating absolutely-integrable functions
    • Integrating on measurable sets
    • Basic properties of the Lebesgue integral
      • Linearity
      • Monotonicity
      • Effects of sets of measure zero
      • Absolute continuty of integration
      • Fatou's Lemma
      • Monotone Convergence Theorem
      • Dominated Convergence Theorem
      • Differentiation w.r.t. a parameter
      • Linear changes of variable
      • Compatibility with Riemann integration
  3. Fubini's Theorem for Rn
  4. L1, L2, and L
    • Completeness
    • Approximation by smooth functions
    • Continuity of translation
  5. Fourier transform on Rn
    • Convolutions
    • Basic properties of Fourier transforms
      • Composition with translation, dilation, inversion, differentiation, convolution, etc.
      • Regularity of transformed functions
      • Riemann-Lebesgue Lemma
    • Inversion Theorem for L1
    • Schwartz class
    • Fourier-Plancherel Transform on L2
      • Its inversion
      • Isomorphism
    • Fourier series
      • Dirichlet and Fejér kernels
      • L2 convergence
      • Pointwise convergence
      • Convergence of Cesàro means

Additional topics

Courses for which this course is prerequisite