Difference between revisions of "Math 641: Functions of a Real Variable"
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#* Completeness of L<sup>p</sup> | #* Completeness of L<sup>p</sup> | ||
#* Density of C<sub>c</sub> in L<sup>p</sup> | #* Density of C<sub>c</sub> in L<sup>p</sup> | ||
| + | #* Duality of the L<sup>p</sup> spaces | ||
#* Convergence in norm | #* Convergence in norm | ||
#* Almost-everywhere convergence | #* Almost-everywhere convergence | ||
Revision as of 11:29, 14 August 2008
Contents
Catalog Information
Title
Functions of Real and Complex Variables 1.
Credit Hours
3
Prerequisite
Math 542 or instructor's consent
Description
Fundamentals of measure and integration, Borel measures, product measures, L^ spaces, introduction to functional analysis, Radon Nikodym theorem, differentiation theory, Fourier transforms.
Desired Learning Outcomes
Prerequisites
Minimal learning outcomes
- Abstract measure theory
- σ-algebras
- Measures
- Positive measures
- Signed measures
- σ-finite measures
- Complete measures
- Measurable spaces
- Measure spaces
- Abstract integration theory
- Abstract measurable mappings
- Measurable real- and extended-real-valued functions
- Integrating simple functions
- Integrating nonnegative functions
- Integrating L1 functions
- Integration on a measurable set
- Measures defined through integration
- Absolute continuity of integration
- Linearity of integration
- Monotone Convergence Theorem
- Fatou's Lemma
- Dominated Convergence Theorem
- Effect of sets of measure zero
- Operations on measures
- Absolutely continuous measures
- Mutually singular measures
- Lebesgue Decomposition Theorem
- Radon-Nikodym Theorem
- Hahn Decomposition Theorem
- Jordan Decomposition Theorem
- Lp spaces
- Hölder's Inequality
- Minkowski's Inequality
- Completeness of Lp
- Density of Cc in Lp
- Duality of the Lp spaces
- Convergence in norm
- Almost-everywhere convergence
- Almost-uniform convergence
- Convergence in measure
- Egoroff's Theorem
- Measures on product spaces
- Tonelli Theorem
- Fubini Theorem
- Measures on topological spaces
- Borel σ-algebra
- Locally compact Hausdorff spaces
- Urysohn's Lemma
- Partitions of unity
- Borel measures
- Locally finite measures
- Regular measures
- Radon measures
- Riesz Representation Theorem (for positive linear functionals on Cc)
- Lusin's Theorem
- Lebesgue measure on Rn
- Existence
- Composition with affine maps
- Differentiation on R and integration
- Derivative of integral is the integrand a.e.
- Functions of bounded variation
- Absolutely continuous functions
- Integrating derivatives of absolutely continuous functions
