Math 241B (Functional Analysis)

Course Topics: Functional Analysis
Instructor: Prof. Michael Holst (5739 AP&M,
Term: Winter 2009
Lecture: 9:30a-10:50a Tu-Th, APM 7421
Class webpage:
  • J. Conway, A Course in Functional Analysis, Springer-Verlag, 2000, 2nd edition.

CATALOG DESCRIPTION: 241A-B. Functional Analysis (4-4)
Metric spaces and contraction mapping theorem; closed graph theorem; uniform boundedness principle; Hahn-Banach theorem; representation of continuous linear functionals; conjugate space, weak topologies; extreme points; Krein-Milman theorem; fixed-point theorems; Riesz convexity theorem; Banach algebras. Prerequisites: Math.240A-B-C or consent of instructor.

GRADES, EXAMS, DATES: Your grade in the course is based on attending the lectures and participating in the class.

ANNOUNCEMENTS, NOTES, ETC: Additional books, papers, and notes that I lectured from throughout the quarter:

SCHEDULE FOR THE LECTURE TOPICS: The overall plan is to develop some remaining topics in linear functional analysis that were not yet covered in 241A, and then to develop some of the fundamental ideas in nonlinear functional analysis. We will motivate almost everything we do by specific examples coming from linear and nonlinear PDE applications. This is very appropriate, since functional analysis as a field has developed in large part as a generalization of tools created for analyzing PDE and integral equations.

The outline below will be updated as the quarter proceeds, based on how the interests of the participants and the lecturer evolve. The first four weeks we cover foundations of linear functional analysis with applications to linear PDE; the last six weeks we focus on developing the basic results of nonlinear functional analysis, with applications to nonlinear PDE.
Lectures Topics Covered
Week 1 (Linear Spaces and Linear Operators)
Quick review of the main topics in linear functional analysis (241A): Sets, fields, topological spaces, metric spaces, linear (vector) spaces. Topological vector spaces, functions, continuity. Normed spaces, inner-product spaces. Cauchy sequences, convergence, closed sets, complete spaces, completion of metric spaces, Banach and Hilbert spaces, Cauchy-Schwarz and triangle inequalities, orthogonality, Parallelogram law, Pythagorean formula. Linear operators on Banach and Hilbert spaces; norms of operators; (Hilbert) adjoint operator. Mappings of spaces, the four fundamental subspaces. Continuity and boundedness of linear operators. Continuity of the norm and inner-product as functions. Bounded and unbounded operators; examples. Compact operators, projection operators. Linear functionals on Banach and Hilbert spaces, the dual norm, the dual space as a Banach space. The double dual, reflexive and separable Banach and Hilbert spaces. Weak/strong convergence of sequences, strong implies weak. Weak-* convergence. Weakly closed, weakly compact, weakly continuous. Strong, weak, and weak-* topologies of a topological vector space. Weakly convergent subsequences extracted from bounded sequences in a reflexive Banach space.
Week 2 (The Three Principles of Linear Analysis and All That)
The Riesz Representation Theorem (RRT). The Hahn-Banach Theorem (Principle I). Open Mapping (Banach-Schauder) and Closed Graph Theorems (Principle II). Principle of Uniform Boundedness (Banach-Steinhaus Theorem; Principle III). Linear and bilinear forms/functionals; continuity (boundedness), coercivity, symmetry, positivity. The Equality of Forms Theorem. The Bounded Operator Theorem. The Lax-Milgram Theorem and proof via the CMT (proof of CMT comes in week 5). The Lions-Stampachia Theorem and relation to Lax-Milgram and RRT. Closed subspaces of a Hilbert or Banach space, the Orthogonal Complement Theorem; Convex sets, the Best Approximation Theorem, Hilbert Space Projection Theorem, the Krein-Milman Theorem.
Week 3 (Linear PDE Operators on Banach Spaces: The Lp, Sobolev, and Besov Classes)
Motivating application for developing tools in linear and nonlinear functional analysis: A second-order quasi-linear boundary-value problem (BVP). Weak formulation of the BVP, the natural appearance of L2 and the L2-based Sobolev space H1 via the Cauchy-Schwarz inequality; Lp spaces on bounded open sets in Rn, the Lp-based Sobolev spaces W[k,p] using the notion of weak derivative. Lp and W[k,p] as Banach spaces (Reisz-Fischer Theorem for Lp and analogue for W[k,p]). Alternative construction of H[k,p] through completion of Coo in the W[k,p] norm. W=H (Meyers-Serin Theorem). Linear operators on Lp; Operator interpolation; Riesz(-Thorin) Convexity Theorem. The Trace Theorem in the Sobolev spaces W[k,p] and use in formulating and solving Dirichlet problems; The Poincare inequality.
Week 4 (Properties of Sobolev Classes; Applications)
The Sobolev Embedding (Imbedding), Compactness, Density, and Extension Theorems for W[k,p], and special cases of Hilbert spaces when p=2. The Sobolev spaces as a Banach Algebra; general Banach Algebras and Schauder Rings. A side comment (connection to numerical analysis): General approximation theorems for Petrov-Galerkin methods for abstract linear operator equations in Banach spaces, Cea's Lemma as a special case. Linear case to get us going on USING functional analysis techniques to prove results for PDE: Well-posedness (existence, uniqueness, stability) of a general linear second order elliptic problem on an open set in Rn using the Lax-Milgram Theorem and the various supporting results we now have on Sobolev spaces.
Week 5 (Nonlinear Maps)
Nonlinear maps on metric spaces and on Banach spaces; Holder, Lipschitz, contraction conditions on maps; the Contraction Mapping Theorem (CMT) in metric spaces and in normed spaces. First variation, Gateaux-variation/differential/derivative, Frechet-variation/differential/derivative. Continuity, equivalent definitions, uniform continuity. Euler conditions for stationarity of a nonlinear functional; necessary and sufficient conditions for optimality. Generalized Taylor expansion. Newton's Method for nonlinear operator equations on Banach spaces; Convergence properties of Newton's method. A general Variational problem for a nonlinear functional on a Banach space (P0), the condition for stationarity as a nonlinear PDE problem (P1), and the strong formulation of the nonlinear PDE problem (P2). Exploiting the connection between problems P0-P2 in Newton's method to enlarge basin of attraction; Galerkin methods for problems P0-P2. Two examples of P0-P2: Example 1: A nonlinear energy functional giving rise to a model scalar semi-linear PDE as condition for stationarity; Example 2: A nonlinear energy giving rise to the nonlinear Hamiltonian constraint in general relativity.
Week 6 (Variational Methods)
Stationary points of functionals on Banach spaces; necessary and sufficient conditions for stationarity (again, but now made rigorous). Proper functionals, coercive functionals, bounded-from-below functions. Convex functionals; lower-semi-continuity; closed sets. Weakly closed sets; weak-lower-semi-continuity; reflexive Banach spaces and the extraction of weakly convergent subsequences. Main theorems on existence of minimizers. Proper formulation of the nonlinear PDE in Example 1: use of Sobolev imbeddings, Holder-inequalities, and Trace inequalities to select the appropriate functions spaces that make the variational and Euler condition problems at least well-defined to start looking for solutions. Well-posedness (existence, uniqueness, stability) of the nonlinear model problem (Example 1) using variational methods. The problem with stronger nonlinearity: the Sobolev critical exponent for space dimension n, and how the variational argument has to be adjusted.
Week 7 (Brouwer Theorem, Energy Estimates, Galerkin Method)
The Brouwer-Fixed Point Theorem and its short proof via the no-Differentiable Retraction Theorem and the determinant lemma, as an alternative to the development of Degree Theory. Well-posedness (existence, uniqueness, stability) of the nonlinear model problem (Example 1) using the Galerkin method and energy estimates, along with the Brouwer Theorem.
Week 8 (Ordered Banach Spaces, Monotone Increasing Maps)
A tutorial on abstract ordered Banach spaces; the maximum principle for abstract operator equations, and the case of elliptic operators. The theory of monotone increasing maps in ordered Banach spaces; statement of the Schauder Fixed-Point Theorem (proof will be given in the final lecture). Well-posedness (existence, uniqueness, stability) of the nonlinear model problem (Example 1) using the monotone increasing maps-based fixed-point arguments (the method of sub- and super-solutions), and a second proof using compactness-based fixed-point arguments (the Schauder Theorem). Construction of compatible barriers (sub- and super-solutions).
Week 9 (Extended Example: Three Different Proofs of Well-Posedness of a Critical Exponent Problem)
The coupled Hamiltonian and momentum constraints in the Einstein equations. Well-posedness of the Momentum constraint using Riesz-Schauder Theory: generalization of Lax-Milgram for operators satisfying Garding rather than coercivity, allows for Fredholm-like "uniqueness implies existence" arguments. Well-posedness of the Hamiltonian constraint using barriers in ordered Banach spaces combined with either monotone increasing maps-based fixed-point arguments, or Variational methods, or compactness-based fixed-point arguments.
Week 10 (The Schauder Theorem: Proof and Application)
Overview of the Schauder-based existence result of Holst, Nagy, and Tsogtgerel of non-CMC solutions to the coupled Einstein constraints. Reduction of the existence result to global barrier construction and the Schauder Theorem. Construction of global barriers (sub- and super-solutions) for the Hamiltonian constraint by scaling solutions to Yamabe-type problems. Proof of the Schauder Theorem through the Approximation Lemma and the Brouwer Theorem.