
Math 241B (Functional Analysis)
Course Topics: Functional Analysis
Instructor: Prof. Michael Holst
(5739 AP&M, mholst@math.ucsd.edu)
Term: Winter 2009
Lecture: 9:30a10:50a TuTh, APM 7421
Class webpage:
http://ccom.ucsd.edu/~mholst/teaching/ucsd/241b_w09
Textbook(s):
 J. Conway,
A Course in Functional Analysis,
SpringerVerlag, 2000, 2nd edition.
CATALOG DESCRIPTION:
241AB. Functional Analysis (44)
Metric spaces and contraction mapping theorem; closed graph theorem;
uniform boundedness principle; HahnBanach theorem;
representation of continuous linear functionals;
conjugate space, weak topologies;
extreme points;
KreinMilman theorem;
fixedpoint theorems;
Riesz convexity theorem;
Banach algebras.
Prerequisites: Math.240ABC 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:
 E. Zeidler,
Nonlinear Functional Analysis and its Applications,
SpringerVerlag, 1991.
 S. Kesavan,
Topics in Functional Analysis and Applications,
John Wiley and Sons, 1998.
 D. Mitrovic and D. Zubrinic,
Fundamentals of Applied Functional Analysis,
John Wiley and Sons, 1998.
 K. Yosida,
Functional Analysis,
SpringerVerlag, 1980.
 M. Struwe,
Variational Methods,
SpringerVerlag, 2000, Third Edition.
 M. Holst, G. Nagy, and G. Tsogtgerel,
Rough Solutions of the Einstein Constraints
on closed manifolds without nearCMC conditions.
Comm. Math. Phys., Vol. 288 (June 2009), No. 2, pp. 547613.
(Paper at arXiv:0712.0798 [grqc])
(Paper at CMP)
 M. Holst, G. Nagy, and G. Tsogtgerel,
Farfromconstant mean curvature solutions of Einstein's constraint
equations with positive Yamabe metrics.
Physical Review Letters, Vol. 100 (2008), No. 16, pp. 161101.1161101.4.
(Paper at arXiv:0802.1031 [grqc])
(Paper at APS)
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, innerproduct spaces.
Cauchy sequences, convergence, closed sets, complete spaces,
completion of metric spaces, Banach and Hilbert spaces,
CauchySchwarz 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 innerproduct 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 HahnBanach Theorem (Principle I).
Open Mapping (BanachSchauder) and Closed Graph Theorems
(Principle II).
Principle of Uniform Boundedness (BanachSteinhaus Theorem;
Principle III).
Linear and bilinear forms/functionals;
continuity (boundedness), coercivity, symmetry, positivity.
The Equality of Forms Theorem.
The Bounded Operator Theorem.
The LaxMilgram Theorem and proof via the CMT
(proof of CMT comes in week 5).
The LionsStampachia Theorem and relation to
LaxMilgram 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 KreinMilman 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 secondorder quasilinear boundaryvalue problem (BVP).
Weak formulation of the BVP,
the natural appearance of L2 and the L2based Sobolev
space H1 via the CauchySchwarz inequality;
Lp spaces on bounded open sets in Rn, the Lpbased Sobolev
spaces W[k,p] using the notion of weak derivative.
Lp and W[k,p] as Banach spaces (ReiszFischer 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 (MeyersSerin 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 PetrovGalerkin 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:
Wellposedness (existence, uniqueness, stability) of a
general linear second order elliptic problem on an
open set in Rn using the LaxMilgram 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, Gateauxvariation/differential/derivative,
Frechetvariation/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 P0P2 in
Newton's method to enlarge basin of attraction;
Galerkin methods for problems P0P2.
Two examples of P0P2:
Example 1: A nonlinear energy functional giving
rise to a model scalar semilinear 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, boundedfrombelow
functions.
Convex functionals; lowersemicontinuity; closed sets.
Weakly closed sets; weaklowersemicontinuity; 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, Holderinequalities, and
Trace inequalities to select the appropriate
functions spaces that make the variational and Euler
condition problems at least welldefined to start
looking for solutions.
Wellposedness (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 BrouwerFixed Point Theorem and its short proof via
the noDifferentiable Retraction Theorem and the
determinant lemma, as an alternative to the development
of Degree Theory.
Wellposedness (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 FixedPoint
Theorem (proof will be given in the final lecture).
Wellposedness (existence, uniqueness, stability) of the
nonlinear model problem (Example 1) using the monotone
increasing mapsbased fixedpoint arguments (the method
of sub and supersolutions), and a second proof using
compactnessbased fixedpoint arguments (the Schauder Theorem).
Construction of compatible barriers (sub and supersolutions).

Week 9 (Extended Example: Three Different Proofs of WellPosedness of a Critical Exponent Problem) 
The coupled Hamiltonian and momentum constraints in the
Einstein equations.
Wellposedness of the Momentum constraint using
RieszSchauder Theory: generalization of LaxMilgram
for operators satisfying Garding rather than coercivity,
allows for Fredholmlike "uniqueness implies existence"
arguments.
Wellposedness of the Hamiltonian constraint using
barriers in ordered Banach spaces combined with
either monotone increasing mapsbased fixedpoint arguments,
or Variational methods, or compactnessbased fixedpoint
arguments.

Week 10 (The Schauder Theorem: Proof and Application) 
Overview of the Schauderbased existence result of
Holst, Nagy, and Tsogtgerel of nonCMC 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 supersolutions)
for the Hamiltonian constraint by scaling solutions
to Yamabetype problems.
Proof of the Schauder Theorem through the Approximation Lemma
and the Brouwer Theorem.

