Math 240A Home Page (Driver, F2016-S17)

 

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• Final Exam is Wednesday, December 7 from 8:00 -- 11:00 AM in APM 5402. -- please bring a Blue Book to write your exam in.
  

• Tentative Finals Week office hours, 10-12 on Monday and Tuesday of finals week.

Grader:              Jacqueline Warren ( email  / 534-49072) AP&M 6446 *This course does not have a TA per se but only a grader. Please only contact the TA about grading issues.

Homework:    Homework  is assigned, collected, and partially graded regularly. Each homework is due at the beginning of each Friday class unless otherwise instructed.

• Please print your names and student ID numbers on your homework. Please staple together your homework pages.

• No late homework will be accepted unless a written verification of a valid excuse (such as hospitalization, family emergency, religious observance, court appearance, etc.) is provided.
   

Math 240A (Fall 2016)

• Reviewed: limit operations, countability, set theoretic manipulations.

• Covered basic notions of metric and pseudo-metric spaces

• Introduced normed and Banach spaces

• Introduced sums over arbitrary sets and prove the prototypical theorem: DCT, MCT, Fatou's Lemma, Tonneli and Fubini Theorems.

• Introduced Holder's and Minikowski's inequalities and showes l^p - spaces were normed spaces.

• Introduced, elementary sets, algebras and described how to construct finitely additive measure on algebras

• We covered sigma - algebras including the Borel and product sigma-algebras

• Covered the notion of countably additive measures

• Gave special case of Caratheodory's construction of measures.

• Used the measure construction theory to describe all Radon measure on the real line and in particular constructed Lebesgue measure.

• Measurable functions and their properties under limits and other calculus operations.

• Construction of the integral from a measure

• Mentioned the relationship of the Lebesgue integral with the Riemann integrable

• General properties of the integral (Fatou's lemma, monotone convergence, Lebesgue dominated convergence, Tchebychev inequality, Jensen's inequality)

• Multiplicative System Theorem and its many applications

• Product measure and the theorems of Tonelli and Fubini.

• Introduced d-dimensional Lebesgue measure and showed it was translation invariant.

Math 240B (Winter 2017)

• More Multiplicative System Theorem and density results.

• Covered the the d-dimensional change of variables theorem including polar coordinates. 

• Lp spaces and related results like Holder and Minikowski inequalities and the different modes of convergence.

• Developed the basics of Banach spaces (=complete normed spaces) including operator spaces, dual spaces, and their noms.

• Hilbert space theory including projection theorems, orthonormal bases, and adjoints of bounded operators.

• Introduced complex measures

• Covered the notation of absolutely continuous measures and and the Lebesgue-Radon-Nikodym theorem in the absolutely continuous setting.

• The dual of dual of Lp spaces.

• Basics of orthonormal bases on a Hilbert space and the applications to Fourier series including Plancherel theorem.

• Hahn Banach Theorem and some of its consequences.

• Notion of reflexive spaces and integration theory of Banach valued funcitons.

• Differential Calculus on Banach spaces and the Inverse and implicit Function Theorems.

• Basic ODE theory on Banach spaces

• Convolution and smoothing operators

• Proof of the d-dimensional change of variables formula

• More Banach space results: Banach Steinhaus Theorem, Open mapping theorem and the closed graph theorem.

• More general point set topology notions and basic notions of compactness.

• Tychonoff's compactness theorem for countable products of compact metric spaces.

• Arzela-Ascolil compactness theorem.

• Compact Operators, Hilbert Schmidt operators, the spectral theorem for self-adjoint compact operators.

• Weak convergence and sequential weak compactness of the unit ball in Hilbert spaces.

Math 240C (Spring 2017)

• General Tychonoff's Compactness theorem.

•  Jensen's inequalities.

• Signed and complex measures including the Jordan and Hahn decompositions.

• Lebesgue-Radon-Nikodym theorem including the notion of  absolute continuity (abstract differentiation of measures)

• Maximal theorem, Lebesgue sets, and differentiation of measures relative to Lebesgue measure.

• Absolute continuity, the fundamental theorem of calculus, and integration by parts.

• Density and approximation theorems including the use of convolution and the Stone Weierstrass theorem.

• Locally compact Hausdorff spaces, Uryshon's Lemma, Tietze Extension Theorem, Partitions of Unity, Alexandrov's compactification, Uryshon's metrization theorem.

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• Fourier Transform and its properties with basic applications to PDE and Sobolev spaces.

• Riesz Markov Representation theorem for measures.

• Some basic Sobolev space theory.

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Possible further topics

• A little complex analysis.

• Distribution theory and elliptic regularity.

• The Spectral Theorem for bounded self-adjoint operators on a Hilbert space.

• Unbounded operators and the Spectral Theorem for self-adjoint operators.

• Properties of ordinary differential equations.

• Differentiable manifolds.

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Last modified on Thursday, 23 March 2017 02:14 PM.