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

 

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240C New Announcements

• Final Exam = Qual Exam. Math 240 Qual is scheduled for Monday, May 22nd from 9:00 -- 12:00 AM in APM 6402. You do NOT need a Blue Book for this exam.
• The qualifying exam is cumulative over the whole years material!!
• Here is the Spring 2012 Qualifying Exam for Practice:  240qual_s2012_test.pdf
  
• There will be no classes during the qual exam week of May 22 -- May 26, 2017.
• No class Monday, May 29 as well as this is Memorial day,

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.

• Holder's and Minikowski's inequalities for l^p - spaces showing in particular that l^p is a normed space.

• Introduced classes of subsets; 1) elementary sets, 2) algebras and 3) sigma-algebras

• Construction of finitely additive measure on algebras from finitely additive function on an elementary class.

• Borel and product sigma-algebras

• Introduced countably additive measures and covered a 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 integral.

• 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)

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

• L^p spaces and related results like Holder and Minikowski inequalities.

• Covered some different modes of convergence including L^p -convergence, convergence in measure, and almost everywhere convergence.

•  Jensen's inequalities.

• More Multiplicative System Theorem and density results in L^p spaces.

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

• Hahn Banach Theorem and some of its consequences.

• Notion of reflexive spaces.

• Bochner type integration theory of Banach valued functions including the fundamental theorem of calculus.

• Basic linear ODE theory on Banach spaces

• Hilbert space theory including projection theorems, Riesz Theorem, and adjoints of bounded operators.

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

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

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

• The dual of L^p spaces including (L^p)*=L^q.

• Convolution inequalities

Math 240C (Spring 2017)

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

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

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

• Properties of convolution and its applications to smoothing operators.

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

• Some general point set topology definitions and notations

  • Tychonoff's compactness theorem.

    Arzela-Ascolil compactness theorem

    The notion of a compact operator.

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

• Locally compact Hausdorff spaces

  • Uryshon's Lemma

  • The general Stone Weierstrass theorem.

  • Tietze Extension Theorem,

  • Partitions of Unity,

  • Alexandrov's compactification,

  • Uryshon's metrization theorem.

• Riesz Markov Representation theorem for measures.

Possible further topics

• Some basic Sobolev space theory. 

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

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

• Basic non-linear ODE theory on Banach spaces

• 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.