Mathematics 421/510 course topics, Winter/Spring 2017

• January 12: We have finished Section 5.1 and are starting Section 5.2.
  • The spaces L^1(mu), C([0,1]) and C^k([0,1]) are mentioned in Section 5.1 and Exercise 9.
  • We will also use L^p spaces as examples. Next time, we will jump to Sections 6.1 and 6.2 to cover Holder's inequality and the duality of L^p spaces.
  • Today I mentioned a special case of Riesz representation theorem (for compact subsets of R^n). This is covered in Chapter 7, in more generality than we need (for very general topological measure spaces). We will not be studying it in detail for now, but will come back to this special case if we have time later in the course.
• January 30: We have finished Sections 6.1 and 6.2:
  • Section 6.1: parts 6.1-6.8 are important and will be used throughout the rest of the course. It's not necessary to memorize the conditions for equality in Holder's inequality, but you should remember that equality can be attained. It is also unnecessary to remember Propositions 6.9-6.12, but you should have some practice in proving results of this type (as per HW#2).
  • Section 6.2: The main result here is Theorem 6.15 and it's generally not necessary to remember all the intermediate steps. Most measures that we use in analysis are sigma-finite. We will return to reflexivity (Corollary 6.16) in connection with the corresponding abstract theory.
• February 28: We have finished Section 5.2 and are starting Sections 5.3 and 5.4:
  • Section 5.2: We covered everything in this section. We also discussed the connection between sublinear functionals and convex sets (not in the textbook, but the main definitions are outlined in HW3, Q1). We also spent additional time on adjoint (transpose) operators, and on annihilators of subspaces (Section 5.2, Questions 22 and 23 respectively).
  • Sections 5.3 and 5.4: we will do this in a different order and skip parts of the material. My current plan is:
    • Weak and weak-* convergence of sequences, with examples (if you need an additional reference, try Lax, Chapters 10 and 11)
    • Principle of Uniform Boundedness for linear functionals, with applications. (The proof in class followed Conway, Chapter III, Section 14. Folland has a different proof in 5.13. Both Conway and Folland state a more general version, for operators insead of functionals.)
    • Weak sequential compactness
    • Types of convergence of sequences of operators (Update, 03/21: we did not have time for this.)
    • Baire Category Theorem, Open Mapping and Closed Graph Theorems (Folland, Section 5.3)
• March 8: Midterm 2 will cover the course material from the Hahn-Banach theorem and its consequences, through weak and weak* convergence, to current material (Principle of Uniform Boundedness and its consequences). This includes Section 5.2 and parts of Section 5.4 from the textbook, but we also spent additional time on material that was only mentioned briefly in the textbook (transpose operators, convex sets, characterizations of weakly and weak* convergent sequences). For more details and updated references, see above. You do not have to memorize all theorems, definitions, etc., but you should remember the following:
  • Basic definitions (bounded linear functionals and operators, dual space, reflexive spaces, the transpose of an operator).
  • Hahn-Banach theorem, both real (5.6) and complex (5.7)
  • Definitions of weak and weak* convergence of sequences (please also review the examples from class).
  • The Uniform Boundedness Principle for functionals (5.13)
• March 21: Next time we will move on to Hilbert spaces (Folland, Section 5.5). I expect that this will take at least 3 classes. If we have time at the end of the course, we will finish with (a special case of) the Riesz Representation Theorem: the dual of C(X), where X is a compact subset of Rⁿ.