Difference between revisions of "Math 541: Real Analysis"
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Revision as of 15:40, 3 April 2013
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 onesemester course specifically on Lebesgue integration in Euclidean space and Fourier analysis.
Prerequisites
Currently, Math 541 requires a semester of singlevariable real analysis and a semester of multivariable 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 absolutelyintegrable 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
 RiemannLebesgue Lemma
 Inversion Theorem for L^{1}
 Schwartz class
 FourierPlancherel 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.