Facts about Hilbert and Banach spaces
Class diary: As we will not follow the same order as the book, here is an outline of the material covered in each class.
1. Introduction to normed and pre-Hilbert spaces:
August 25: Introduction. Definitions of norm and inner product. Examples of normed and pre-Hilbert spaces. Section I.1 (1.1, 1.2, 1.3, 1.5, 1.7. Example 1.8 has not been seen in class, read it in the book). Section III.1 (1.1, 1.2, 1.8, 1.9. Read also Examples 1.10 and 1.11).
August 27: Inequalities: Cauchy-Schwarz, Hölder's inequality and Minkowsky's ineq for Lp spaces. Characterization of distances arising from a norm. Interplay between the topological and vector structure. Section I.1 (1.4), Section III.1 (1.3)
August 30: Interplay between the topological and vector structure (continued): Continuity of operations, closed subspaces, convex subsets, completeness. Characterization of norms arising from an inner product. Section I.2 (2.3,2.4)
2. Hilbert spaces:
September 1: Orthogonality and projections. Rest of Section I.2.
September 3: Problems about projections
September 8: The Riesz Representation Theorem. All of Section I.3
September 10: Lax-Milgram Theorem and problems about Riesz Representation (not in book).
September 13-17: Orthonormal sets and bases. Gram-Schmidt. Bessel's Inequality. Characterization of a basis (the Main Theorem) Section I.4.
September 20: The Fourier basis in L^2[0,1]. Every Hilbert space has a basis.
September 22: The Haar and Walsh bases in L^2[0,1]. Cardinality of bases. Every Hilbert space is isometrically isomorphic to a space of sequences.
September 24: Exercises for Section I.4.
3. Banach spaces:
September 27: Linear operators on Banach spaces (Section III.2 (Proposition 2.1, Exercise 1). Def of Banach Algebra.
September 29: Equivalence of norms (Section III.1 (1.5), Section III.3 (3.1)). Norms in quotient spaces (Section III.4).
October 1: Norms in quotient spaces (continued). Hahn-Banach and its corollaries (Section III.6). The two versions of the Hahn-Banach theorem are Thm 6.2 and Coroll. 6.4 in the book. Our Corollary 1 is 6.5 in the book, our Corollary 3 is 6.7, our Corollary 4 is 6.8. Our density criterion is Corollary 6.14.
October 4: The dual of l^1 is l^∞. The dual of l^p is l^(p') (The book only mentions it in III.5.9 and 5.10)
October 6: The application J:X->X**. The dual of c0 is l^1. Reflexive spaces. c0, l^1, l^∞ are not reflexive. (Section III.11)
October 8: The dual of L^p(Ω,μ) is L^(p')(Ω,μ),1≤p<∞. (Appendix B in book)
October 11: Banach limits (Section III.7)
October 13: Exercises about the bidual space.
4. The Baire Category Theorem and its consequences
October 15: Baire's Theorem (not in book) and some corollaries.
October 18: Exercises on Baire's Theorem.
October 20: The Open Mapping Theorem (Section III.12). Application to Fourier series in L^1[0,1]
October 22: Fourier series in L^1[0,1]. Definition of closed operators.
October 25: Two examples of closed operators that are not continuous. The Closed Graph Theorem. Separable spaces are isomorphic to a quotient space of l^1.
October 27: Complemented subspaces (III.13). The Principle of Uniform Boundedness (Section III.14)
October 29: The Principle of Uniform Boundedness. Application to Fourier series in L^p[0,1].
November 1: Fourier series (continued)
5. Topological vector spaces and Weak Topologies
November 3: Introduction to weak topologies and topological vector spaces (TVS)
November 5: Locally convex spaces (LCS): Seminorms and their properties.(Section IV.1)
November 8: Examples of LCS. Which LCS are metrizable? (Sections IV.1,IV.2)
November 10: Metrizable LCS continued. Lp(0,1), p<1, is a metric space without a bounded convex open set .(Section IV.2)
November 12: Separation properties of TVS. Normable LCS.