Difference between revisions of "Math 541: Real Analysis"
From MathWiki
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#** Compatibility with Riemann integration | #** Compatibility with Riemann integration | ||
# Fubini's Theorem for <b>R</b><sup>n</sup> | # Fubini's Theorem for <b>R</b><sup>n</sup> | ||
− | # L<sup>1</sup>, L<sup>2</sup>, and L<sup>∞</sup> | + | # L<sup>1</sup>, L<sup>2</sup>, and L<sup>∞</sup> |
#* Completeness | #* Completeness | ||
#* Approximation by smooth functions | #* Approximation by smooth functions | ||
+ | #* Continuity of translation | ||
+ | # Fourier transform on <b>R</b><sup>n</sup> | ||
+ | #* 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<sup>1</sup> | ||
+ | #* Schwartz class | ||
+ | #* Fourier-Plancherel Transform on L<sup>2</sup> | ||
+ | #** Its inversion | ||
+ | #** Isomorphism | ||
+ | |||
</div> | </div> | ||
Revision as of 10:58, 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
Prerequisites
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