# Math 541: Real Analysis

## Contents

## Catalog Information

### Title

Real Analysis.

### Credit Hours

3

### Prerequisite

Math 341; 314 or 342; or equivalents.

### Description

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

## Desired Learning Outcomes

Math 541 is a one-semester course specifically on Lebesgue integration in Euclidean space and Fourier analysis.

### Prerequisites

Currently, Math 541 requires a semester of single-variable real analysis and a semester of multi-variable Calculus. Replacing these prerequisites by Math 342 would imply that the new version of Math 541 could presuppose that students had been exposed to the geometry of **R**^{n} and to metric spaces, which would make it easier to cover the core topics listed below.

### Minimal learning outcomes

Outlined below are topics that all successful Math 541 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

- Lebesgue measure on
**R**^{n}- 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

- Lebesgue integration on
**R**^{n}- 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

- Fubini's Theorem for
**R**^{n} - L
^{1}, L^{2}, and L^{∞}- Completeness
- Approximation by smooth functions
- Continuity of translation

- Fourier transform on
**R**^{n}- Convolutions
- Basic properties of Fourier transforms
- Composition with translation, dilation, inversion, differentiation, convolution, etc.
- Regularity of transformed functions
- Riemann-Lebesgue Lemma

- Inversion Theorem for L
^{1} - Schwartz class
- Fourier-Plancherel Transform on L
^{2}- Its inversion
- Isomorphism

- Fourier series
- Dirichlet and Fejér kernels
- L
^{2}convergence - Pointwise convergence
- Convergence of Cesàro means

### Textbooks

Possible textbooks for this course include (but are not limited to):

### Additional topics

Extra time could be used to go into Fourier analysis in more depth.