Difference between revisions of "Math 541: Real Analysis"
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=== Prerequisites === | === Prerequisites === | ||
| − | Currently, Math 541 requires a semester of single-variable real analysis and a semester of | + | Currently, Math 541 requires a semester of single-variable real analysis and a semester of multi-variable Calculus. Replacing these prerequisites by [[Math 316]] would imply that the new version of Math 541 could presuppose that students had been exposed to the geometry of <b>R</b><sup>n</sup> and to metric spaces, which would make it easier to cover the core topics listed below. |
=== Minimal learning outcomes === | === Minimal learning outcomes === | ||
Revision as of 11:29, 13 August 2008
Contents
Catalog Information
Title
Real Analysis.
Credit Hours
3
Prerequisite
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
Currently, Math 541 requires a semester of single-variable real analysis and a semester of multi-variable Calculus. Replacing these prerequisites by Math 316 would imply that the new version of Math 541 could presuppose that students had been exposed to the geometry of Rn and to metric spaces, which would make it easier to cover the core topics listed below.
Minimal learning outcomes
- 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
- 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
- Fubini's Theorem for Rn
- L1, L2, and L∞
- Completeness
- Approximation by smooth functions
- Continuity of translation
- 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
