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> and normed linear spaces |
| − | #* | + | #* Completeness |
#* Approximation by smooth functions | #* Approximation by smooth functions | ||
</div> | </div> | ||
Revision as of 10:45, 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∞ and normed linear spaces
- Completeness
- Approximation by smooth functions
