AMATH 731  Applied Functional Analysis, Fall 2018
AMATH 731: Applied Functional Analysis (Fall 2018)
Instructor: E.R. Vrscay
Department of Applied Mathematics
University of Waterloo
Description
This is a core course for graduate students in Applied Mathematics. It will
also be of
interest to students in Engineering and Science who wish to understand and use methods
from classical analysis as well as functional analysis. Basic concepts from functional analysis are introduced and illustrated
with applications in various areas such as mechanics, control theory, boundary value problems for PDEs, numerical
analysis and quantum mechanics.
Outline

Complete metric spaces, contraction mapping theorem

Banach spaces, completion, function spaces

Linear operators, inverses, approximate solution to operator equations

Frechet derivatives, NewtonKantorovich method, Schauder fixed point theorem

Hilbert spaces, projections, generalized Fourier series

Riesz representation theorem, Ritz method, generalized solutions,
Sobolev spaces, LaxMilgram theorem

Compact operators, Spectral theorem
Course "textbook"

Applied Functional Analysis, Course notes for AMATH 731, by D. Siegel.
Available from Media.doc Copy Centre, MC 2018.
Supplementary notes will be provided from time to time. They will be posted below. (Previous versions are already posted here  they will be updated
as we progress through the course.)
Recommended References (these books will be placed on oneday reserve at the Davis Library)

Introductory Functional Analysis with Applications, by E. Kreyszig (1978)

Functional Analysis: Applications in Mechanics and Inverse Problems, by
L.P. Lebedev, I.I. Vorovich and G.M.L. Gladwell (1996)

Linear Operator Theory in Engineering and Science, by A.W. Naylor and G.R. Sell (1982)

Applied Functional Analysis, Applications to Mathematical Physics,
by E. Zeidler (1995).

Nonlinear Functional Analysis and its Applications, Vol. 1,, by E. Zeidler (1985)
Instructor: E.R. Vrscay, MC 6326, ext. 35455, ervrscayATuwaterloo.ca
Prerequisite: Advanced calculus, linear algebra and elementary real analysis
Lecture times: 10:00  11:30 a.m., Tuesdays and Thursdays, in MC 6460
(the Applied Math Seminar Room).
First lecture: Tuesday, September 11, 2018 at 10:00 a.m. in MC 6460.
Final examination (written, 3 hour, no aids): Monday, December 10,
2018. 12:303:30 p.m. in MC 4041.
Material covered in lectures to date
(This list will hopefully be updated at the end of each lecture or week.)
Each set of notes will be revised from its 2017 version before being handed out in class this Fall 2018
term. Titles with "(2018)" have been revised or added this term.

Some introductory comments (2018)

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences," by
E. Wigner, Comm. Pure Appl. Math. 13(1), 114 (1960)

Some important results from real analysis (2018)

Revised version of proof of Theorem 2.2, Course Notes, p. 9 (2018)

Addendum to p. 12 of Course Notes  annotated Proofs of Proposition 2.4 (2018)

Open and closed subsets of a metric space: An interesting example (2018)

Addendum to p. 15 of Course Notes (2018)

Supplementary remarks to Section 2.7, ''Initial Value Problem'' (2018)

Excerpt from NaylorSell book on a ''nonlinear filter'' (2018)

A note on pseudometrics (2018)

A metric space for sets (Hausdorff metric for nonempty compact sets) (2018)

Hitchhiker's Guide to "FractalBased" Function Approximation and Image
Compression (ERV), MathTies 1995

Supplementary notes on normed linear spaces (for Section 3.1, p. 25, of Course Notes) (2018)

An alternate proof of Theorem 3.3, Course Notes, p. 26 (2018)

Supplementary notes on compactness (for Section 3.3, p. 2728, of Course Notes) (2018)

Riesz lemma; Unit ball in X compact implies dim(X) finite (pp. 7881 from Kreyszig) (2018)

Additional notes on Theorem 3.10 (Schauder Fixed Point Theorem), p. 30 of Course Notes (2018)

Addendum to p. 36 of Course Notes  Alternate proof of L cts on X => L bounded.
(2018)

Norms of linear operators: A simple example in R^2
(for Section 3.6, p. 36 of Course Notes) (2018)

Addendum to p. 38 of Course Notes  Additional material for Example 3.7: modifying
the norm for the space C1 so that the differentiation operator is continuous.
(2018)

Additional comments on Example 3.8, p. 39. (2018)

Preconditioned Krylov subspace methods (1 lecture by Yangang Chen in
CS 475/675)  to supplement discussion in Section 3.7 on Condition Number (2018)

Additional notes on B(X,Y) and B(X)
(for Section 3.8, p. 43, of Course Notes) (2018)

Banach's fixed point theorem and iterative solutions of systems of
linear equations (for Section 3.8, p. 43, of Course Notes)
not covered this year (2018)

Neumann series solution to f = g + Lf  a simple example.
For Section 3.9, p. 45 of Course Notes. (2018)

From Multivariable Calculus to Gateaux Derivatives to Frechet Derivatives.
As an introduction to Section 3.10, ``Frechet derivative,''
p. 4546, of Course Notes (2018)

Additional notes on Frechet derivatives
(for Section 3.10, p. 4546, of Course Notes) (2018)

Mean value theorems for functions of several variables,
as an introduction to Section 3.11, ``Generalized
Mean Value Theorem,'' of Course Notes (2018)

Dynamics of iteration in R^n toward locally attractive fixed points
(for Section 3.11, p. 4749, of Course Notes) (2018)

Some preliminary notes on the NewtonKantorovich method
(for Section 3.12, p. 4951, of Course Notes) (2018)

Using the NewtonKantorovich method to locate fixed points of a function
(for Section 3.12, p. 4951, of Course Notes) (not
covered this year) (2017)

Newton's method in the complex plane  from ERV's lecture notes
in PMATH 350  to illustrate fractal basin boundaries, for
information only (2018)

Julia sets and MandelbrotLike Sets Associated With Higher Order
Schroder Rational Iteration Functions: A Computer Assisted Study,
E.R. Vrscay, Math. Comp. 46, 151169 (1986).

Supplementary notes (handwritten) on stability of continuous and discrete
dynamical systems.
(for Section 3.13, p. 52, of Course Notes) (2018)

Notes on theory of iterated function systems (mentioned in class  for
your own background reading)

Finishing the proof of Theorem 4.2, p. 57 of Course Notes (2018)

Additional notes on inner product spaces and Hilbert spaces (2018)

Some basics of integral equations (not covered in class, but for
your own background reading if you wish) (2018)

Additional notes on approximations and projections (2018)

A comment on the Riesz Representation Theorem (Course Notes, p. 65) (2018)

Additional notes on adjoint and unitary operators (2018)

Fourier transforms and the ''Sampling Theorem'' (2008) (not covered in class)

Supplementary notes on Compact Linear Operators (2017)

Some computations to determine/estimate eigenvalues of simple compact linear operators (2017)

Supplementary notes on the spectrum of a linear operator (2016) (not covered in class)

Supplementary notes on SturmLiouville problems
(to accompany Section 4.10 of Course Notes) (2017)

Generalized solutions, Sobolev spaces (2017)

Sobolev spaces, weak solutions Part II (2017)

Weak convergence in a Hilbert space and the Ritz approximation method (2014) (not covered in class)

An introduction to multiresolution analysis and wavelets (2008) (not covered in class)

Dirac "delta function" distribution (2008) (not covered in class)