Linear Analysis (Real Analysis III, Introductory Functional
Analysis)
Math 502
Fall 2007
Instructor: Prof. Dr. rer. nat. Henri Schurz
AG 168 on Mon - Wed - Fri at 10 - 11 am
(core-time for 3 credits)
Location: AG 168
Office Hours in 2007: Neckers 265, MWF 11:00-12:55pm
(Last Update: 10/31/07)
Textbook(s):
- [1.] Principles of Real Analysis, 3rd Edition.
C.D. Aliprantis and O. Burkinshaw, Academic Press, San Diego and London,
1998 (hardcover) or 2004 (softcover).
- [2.] Introductory Real Analysis, Rev. English Edition.
A.N. Kolmogorov and S.V. Fomin, Dover Publications, Dover, 1975.
- [3.]
Foundations of Modern Analysis. A. Friedman, Dover, New York, 1982.
- [4.]
An Introduction to Linear Analysis,
1st Edition. H. Schurz, Lecture Notes, Southern Illinois
University, Carbondale, 2007.
Course
Description: This one semester course is a basic
introduction to real linear analysis, functional analysis and operator theory. The emphasis is put to
end up in a capacity to understand linear spaces, linear functionals and
operators, and related aspects such as spectral theory on Hilbert spaces.
Its purpose is to develop many of the advanced mathematical tools that are
necessary for the understanding of all other advanced courses in analysis.
Prerequisite is strong grasp of topics of Math 452 and 501.
-
This course is an introduction to analysis in linear infinite-dimensional
spaces. Its purpose is to introduce function spaces that are used in the
formulation of modern mathematical models in economics, the sciences, and
engineering involving topics such as control theory, partial differential
equations, and probability. Topics will include: Banach spaces, the
Hahn-Banach Theorem, the uniform boundedness principle, the closed-graph
theorem, the open-mapping theorem, weak convergence, reflexive and separable
spaces, adjoint operators, Hilbert spaces, and the Riesz representation
theorem, among many more topics.
We will hopefully find a student-friendly adapted way of teaching.
The course is directed to graduate students, hence to mathematically
more advanced participants, however newcomers which bring their
unbounded willingness to learn new mathematical techniques are welcomed as well.
The almost complete manuscript of text [4] is available in my office
(and being developed as course advances).
Real and linear analysis is certainly a very large and active field, and we will certainly
not be able to cover all of the important techniques in a
one-semester course, so I intend to let the interests and needs of the
registered students guide the choice of some more specific topics to be studied
to some extent.
(I am already afraid of skipping very important issues.)
Course
History: This course is taught at nearly all
universities throughout the world and considered to be the "gate to
modern analysis". There are plenty of books related to measure,
infinite-dimensional and operator theory.
Thus, stay with me and learn about more recent developments.
Prerequisites
and Development of Contents:
This course should be accessible to any student with a strong grasp
of theory of single variable calculus and some prelude of vector analysis.
Officially prerequisite is Math 501. However, the main thing is you
are willing to learn with me.
The content of this course itself should very nearly coincide with
my new manuscript in progress, although from time to time I will
have to give some more details from other standard references.
I guess that you haven't seen the concepts developed in the way that they
are here. Thus, for your pleasure and convenience, I will give
a concise and relatively selfcontained summary whatever is needed to
understand the related course material. The plan is for us to cover most of
the textbook, as time permits. Everything will be somewhat experimental, and I
hope the audience will forgive me that I am not perfect. However, be sure that I
will do my very best to please you and your expectations. My
advantage is that I am very enthusiastic such that this "fire" can
carry over to you and above all I am clearly very organized as reviewers
write on my style.
Readings,
Problem Sets, Exams:
Readings are from textbook(s), and problem sets will be from the texts
I hand out in the lecture and from my manuscript, and
it will be assigned in class and additionally published at my
homepage. Exams will cover all material covered in the lectures and/or
the readings. I hope to have put a fair effort into his
presentation from very practically oriented point of view,
and my approach is probably quite different from what you've seen
before (hopefully not). I therefore encourage you to read the textbook and
related material,
and even better read extra literature for those they want to have
exp(A+) grade at the very end with me.
Exam
Dates:
- Midterm: 4:00-6:00pm, before October 15, 2007 (as class room exam in Neckers 156, extra scheduled, 2hr)
- Project: December 8, 2007 (written document, orally defended)
- Final exam: between December 8 - December 14, 2006 (as oral exam, 1hr)
My Grading Strategies - Grade
Distribution: Grading relative to the best attained score, no
curve fitting, no homework make-ups, no quizz make-ups!
- 25% Homeworks and Quizzes, 25% Project, 25% Midterm, 25% Final Exam
- A+ >= 96%, A >= 93%, A- >= 90% (absolute minimum)
- B+ >= 85%, B >= 80%, B- >= 75% (absolute minimum)
- C+ >= 70%, C >= 65%, C- >= 60-55% (absolute minimum)
- The project is mandatory which consists of working on a
theoretical topic related to real and linear analysis, measure and integration theory
(such as the construction of nonmeasurable sets, martingale theory,
Banach space of functions of bounded p-variation, signed measures, etc.)
I expect that we periodically meet
to monitor the progress of the project work (at least one time
per month). It is individually done and needs to be
turned in as a written document and defended in an oral academic
presentation by Dec 8.
Course
Syllabus in PS Format (Old, Last Update: 01/18/05) - This is in postscript format, you
will need ghostview to read it!
Please, note that GHOSTVIEW is required to read the postscript files after
downloading! See http://www.cs.wisc.edu/~ghost/index.html for more software
product information.
Course
Syllabus in PDF Format (Old, Last Update: 01/18/05) - This is in pdf format, you
will need XPDF or ACROBAT readers to read it!
Assignment 1 (ps-file) or
(pdf-file)
(Due 09/26/07)
Assignment 2 (ps-file) or
(pdf-file)
(Due 10/26/07)
Assignment 3 (ps-file) or
(pdf-file)
(Due 11/16/07) -
Assignment 4 (ps-file) or
(pdf-file)
(Due 12/06/07) -
This is in postscript and pdf format, you
will need GHOSTVIEW to read postscript and ACROBAT READER to read pdf,
resp.!
Current Course Outline
(Need to be Revised, Last Update: 10/31/07):
Week 1 : I. Linear Spaces: Preliminaries: Literature, linear, vector and metric spaces,
basic operations, isomorphic spaces, dimension, infinite-dimensional spaces,
basis, linear dependence and independence, vector sum, vector product with scalars,
normed linear spaces, metric linear spaces, generalized triangle inequality,
Frechet spaces, Banach spaces, completeness, sequences, series, sequence of
partial sums, convergence, simple convergence, absolute convergence,
direct product, equivalence of norms, convex subsets,
Subspaces, Linear Hull and factor Spaces: (linear) subspace, linear hull,
coset, residue class, factor space, codimension
Week 2 : partial and total orderings, Zorns lemma, Hamel basis, (relative)
compactness, separable spaces, Characterization of Finite-dimensional
Spaces: Riesz theorem of almost vertical, closedness of subspaces,
compactness and finite-dimensionality, equivalence of all norms,
elliptic, max, sum and Euclidean norms, example of unit balls
Week 3 : Introduction to Hilbert Spaces: bilinear form, scalar product,
fundamental properties, unitary spaces, notion of real and complex Hilbert
space, Schwarz inequality, continuity of scalar product,
projection theorems I + II,
orthogonal systems, orthonormal systems (ONS), orthogonal complement,
parallelogram law, Pythagoras theorem,
Week 4 :
Gram-Schmidt orthogonalization procedure,
Fourier coefficients, Fourier series, examples of ONS (Fourier, Legendre,
Hermite, Laguerre, Jacobi, Chebyshev), Fourier decomposition, direct
orthogonal sum, Bessel inequality, convergence characterization of
series in Hilbert spaces, Parseval identity
Week 5 :
Further General Inequalities in Banach Spaces: Youngs, Hoelder, Generalized
Hoelder, Minkowski, Generalized Minkowski,
Lyapunov, Jensens inequalities,
Fixed Point Principles: Contractions, nonexpansive and Lipschizian mappings,
defect inequality, uniform continuity of Lipschitzian mappings, Banach's
contraction mapping principle (CMP), method of successive approximation
Week 6 : Proof of Banach's CMP, application to one-sided Lipschitz ODEs,
existence and uniqueness of global solutions of IVPs, Shilovs closedness
of fixed points, generalized Banach's CMP, Riedrichs FPT, Weissingers
FPT, Application to existence of unique local solutions of IVPs,
Picard-Lindeloef theorem, generalized contractions, Krasnoselskii's FPT
Week 7 :
Istratescu's FPT of nonexpansive mappings, Kakutanis FPT (1941),
Zeidler's concept of retractions and retracts, Brouwer's FPT (1912),
example of Kakutani (1943), Kakutanis maspping, approximation operator
for compact mappings, epsilon-nets and compactness, compact mappings,
Schauders FPTs (1927)
Week 8 : II. Major Theorems of Linear and Functional Analysis:
Linear Operators and Linear Functionals: Basic definitions, commutative and
noncommutative operators, operator products, operator sums, operator norm,
characterization theorems of continuous linear operators, linear spaces
C^0(V1,V2) and L(V1,V2), uniform convergence
Week 9 : Banach space C^0(V1,V2), concept of Banach algebras,
further characterization of bounded linear operators, Banach-Steinhaus
Theorem, principle of uniform boundedness, strong convergence, Baires lemma
(Baires category theorem)
Week 10 :
Inverse operators, invertibility, linearity of inverses, boundedness of
inverses, series representation, continuity of inverses,
Banachs open mapping theorem, Banachs closed graph theorem, derivative
operator as unbounded operator
Week 11 :
Linear Functionals and Transforms: Concept of a linear functional,
continuity, linearity, Dirac functional, projection functional, norm
functional, subadditivity, homogeneity, conjugate-linear functionals,
null space of functionals, representation theorem using linear functionals,
subspace character of null spaces, Minkowski functional, convex functionals
Week 12 : Major Theorems of Functional Analysis:
Hahn-Banach-Theorem, separation theorems, Riesz Theorem, applications to
PDEs, dual spaces, reflexive spaces
Week 13 :
Adjoint operators, self-adjoint operators, bounded operators on Hilbert spaces,
spectral theory and positive operators, spectral representation of
self-adjoint operators, unbounded operators
Week 14 : Thanksgiving Break
Week 15 : Applications to Volterra Equations, Integral Equations of
Fredholm-type, Interpolation, Approximation Theory, Energy Operator,
Neumann Series
Week 16 : Project Presentations, Stone-Weierstrass Theorem, Arzela-Ascoli
Theorem, Tychonoffs Theorem
Further Interesting Related Topics / Miscallanea .....
Read this
Without Tears:
WHAT IS EXPECTED OF YOU
(From "Teaching at the University Level" by Stephen Zucker, Notices Amer.
Math. Soc. 43 (1996), p. 863))
25 Further
Introductory Readings To Real Analysis, Measure and Integration -Theory
For The BEYOND-THE-HORIZON Student (optional, for
those they want to have profound readings I recommend to work through this
list, and make your personal preferences, but do not expect to understand
them all immediately):
1. M. Adams and V. Guillemin: Measure Theory and Probability, Birkhaeuser,
Boston, 1996 [Probabilistic Touch].
2. R.G. Bartle: The Elements of Integration and Lebesgue Measure, Wiley, New
York, 1966 [Jumps Immediately into Measurable Functions and Real
Calculations, Late Use of Set Algebra Concepts].
3. H. Bauer: Mass- und Integrationstheorie, de Gruyter, Berlin, 1992 (German, 2nd edition)
[Good Easy and Compact Reading].
4. J. Bellach, P. Franken, E. Warmuth and W. Warmuth: Mass und Integral und
Bedingter Erwartungswert, Unknown Binding, Berlin, 1978 (German) [Probabilistic Touch].
5. P. Billingsley: Probability and Measure, Wiley, New York, 1986
[Very Probabilistic Touch]
6. J.L. Doob: Measure Theory, Springer-Verlag, New York, 1994 [Classic].
7. J. Elstrodt: Mass- und Integrationstheorie, Springer-Verlag, Heidelberg,
1996 (German) [Interesting].
8. G.B. Folland: Real Analysis: Modern Techniques and Their Applications,
Wiley, New York, 1999 (2nd edition) [State-of-the-Art].
9. H.L. Royden, Real Analysis, 3rd Edition, Macmillan, New York, 1988
[Basics of Real Analysis and Measure Theory, But Some Depth Missing].
10. N.B. Haaser and J.A. Sullivan: Real Analysis, Dover, New York, 1991 [Easy to
Read].
11. P.R. Halmos: Measure Theory, Springer-Verlag, Berlin, 1950 [Classic].
12. H. Koenig: Measure and Integration: An Advanced Course in Basic Procedures
and Applications, Springer-Verlag, Berlin, 1997 [Not So Easy Reading, State-of-the-Art].
13. A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis, Dover, New York, 1975
[Comprehensive, Elementary Introduction, Well-Structured]
14. A.N. Kolmogorov and S.V. Fomin: Measure, Lebesgue Integrals and Hilbert
Space, New York / London, 1961 [Functional-Analytic].
15. H. Michel: Mass und Integrationstheorie, Deutscher Verlag der
Wissenschaften, Berlin, 1978 (German) [Interesting].
16. I.P. Natanson: Theorie der Funktion einer reellen Veraenderlichen, Verlag
Harri Deutsch Thun, Frankfurt/Main, 1981 (German, Original from
Akademie-Verlag, Berlin, 1975) [Excellent, Elementary, Easy to Read].
17. W. Rudin: Principles of Mathematical Analysis, McGraw-Hill, St. Louis, 1976
[Real Analysis I: Standard Introduction to Real Numbers and Real Functions]
18. W. Rudin: Real and Complex Analysis, McGraw-Hill, St. Louis, 1987 [Good
Continuation of Standard Material, Extension to Complex Case]
19. W. Rudin: Functional Analysis, McGraw-Hill, St. Louis, 1991 [Elements of Real Analysis
III: Banach Algebras, Spectral Theory and Fourier Transforms]
20. G.P. Tolstow: Mass und Integral, Akademie Verlag, Berlin, 1991 (German)
[Easy to Read]
21. H. Triebel: Analysis and Mathematical Physics, Reidel Publ. Co, Dordrecht,
1986 (Original From Teubner, Leipzig, 1986) [A Must for Any Mathematician - Biblic Character].
22. A.C. Zaanen: Linear Analysis: Measure and Integral, Banach and Hilbert
Space, Linear Integral Equations, Unknown Binding [Only Used Available].
23. A.C. Zaanen: Introduction to the Theory of Integration, North-Holland Pub.
Co, Amsterdam, 1958 [Classic Integration].
24. B.R. Gelbaum and J.M.H. Olmsted: Counterexamples in Analysis, Dover, Mew York, 2003 (ISBN
0-486-42875-3) [Real Number Examples].
25. K.J. Arrow and M.D. Intriligator (eds): Handbook of Mathematical Economy, Vol.
1, North-Holland, Amsterdam, 1981 [Applications of Measures and Concepts of
Metric Spaces to Economics].
40 Further
Introductory Readings To Measure-Theory-Based Probability and Stochastic Processes
For The BEYOND-THE-HORIZON Student (optional, for
those they want to have profound readings in measure-axiomatic-based
probability theory, I recommend to work through this
list, and make your personal preferences, but do not expect to understand
them all immediately, it took me more than 18 years, and we are still working
on it!):
L. Arnold: Stochastic Differential Equations: Theory and Applications,
Krieger Publishing Company, Malabar (FL), 1992
(reprinted, Wiley, New York, 1974, German original, Oldenburg Verlag, 1973).
L. Arnold: Stochastic Dynamical Systems, Springer, Berlin, 1998.
H. Bauer: Wahrscheinlichkeitstheorie, deGruyter, Berlin, 1991 (English
translation, 1996).
A.T. Bharucha-Reid: Elements of the Theory of Markov Processes and
Their Applications, Dover, Minneola (NY), 1997 (ISBN 0486695395).
A.A. Borovkov: Wahrscheinlichkeitstheorie, Akademie-Verlag, Berlin, 1976.
P. Bremaud: Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues,
Texts in Applied Mathematics 31, Springer, New York, 1999 (ISBN 0387985093).
M. Capinski and E. Kopp: Measure, Integral and Probability, Springer,
New York, 1999 (ISBN: 3540762604).
K.L. Chung: A Course in Probability Theory Revised (3rd edition),
Academic Press, London, 2001 (ISBN: 0121741516).
J.L. Doob: Stochastic Processes, Wiley, New York, 1953.
R. Durrett: Probability: Theory and Examples, Duxbury Press, Belmont (CA),
1995.
E.B. Dynkin: Markov Processes I, II, Springer, Berlin, 1965 (Russian
original, Fizmatgiz, Moscow, 1963).
W. Feller: An Introduction to Probability and Its Applications II, Wiley,
New York, 1971.
T. Gard: Stochastic Differential Equations, Marcel Dekker, Basel, 1988.
I.I. Gikhman and A.V. Skorochod: Introduction to the Theory of Random
Processes, Dover, Minneola (NY), 1996 (translation of Russian original,
Nauka, Moscow, 1965).
B.V. Gnedenko: The Theory of Probability (in Russian), Mir, Moscow,
1988.
I.A. Ibragimov and Yu.V. Linnik: Independent and Stationary Sequences
of Random Variables, Addison-Wesley, Reading, 1968.
I.A. Ibragimov and Yu.A. Rozanov: Gaussian Random Processes, Springer,
New York, 1978.
J. Jacod and A.N. Shiryaev: Limit Theorems for Stochastic Processes,
Springer, New York, 1987.
F. Jones: Lebesgue Integration on Euclidean Space (Revised Ed.),
Jones & Bartlett Pub., 2000 (ISBN 0763717088).
D. Kannan: An Introduction to Stochastic Processes, North-Holland, New York,
1979.
I. Karatzas and S.E. Shreve: Brownian Motion and Stochastic Calculus,
Springer, New York, 1991.
S. Karlin and H.M. Taylor: A First Course in Stochastic Processes,
Academic Press, New York, 1975; A Second Course in Stochastic Processes,
Academic Press, New York, 1981.
R.Z. Khas'minskii: Stochastic Stability of Differential Equations, Sijthoff
& Noordhoff, Alphen aan den Rijn, 1980 (translation of Russian original,
1969).
A.N. Kolmogorov: Grundbegriffe der Wahrscheinlichkeitsrechnung,
Springer, Berlin, 1933 (Reprint, 1973); Foundations of the Theory of
Probability, Chelsea, New York, 1956.
N.V. Krylov: Introduction to the Theory of Diffusion Processes,
AMS, Providence, 1996 (translation of Russian original, 1989).
J. Lamperti: Stochastic Processes, Springer, New York, 1977.
G.F. Lawler: Introduction to Stochastic Processes, Chapman & Hall
Probability Series, CRC Press, New York, 1995 (ISBN 0412995115).
P. Levy: Processus Stochastiques et Mouvement Brownien,
Gauthier-Villars, Paris, 1948.
R.Sh. Liptser and A.N. Shiryaev: Theory of Martingales, Kluwer,
Dordrecht, 1989.
V.V. Petrov: Sums of Independent Random Variables, Springer, Berlin, 1975;
Limit Theorems for Sums of Independent Random Variables (in Russian), Nauka,
Moscow, 1987.
Yu.A. Prokhorov and Yu. Rozanov: Probability Theory, Springer, New
York, 1969.
P. Protter: Stochastic Integration and Differential Equations,
Springer, New York, 1990.
A. Renyi: Probability Theory (in Hungarian, German Translation available),
1954.
S.I. Resnick: Adventures in Stochastic Processes, Birkhauser, Boston,
1992.
S.I. Resnick: A Probability Path, Springer, New York, 1998
(ISBN 081764055X).
D. Revuz and M. Yor: Continuous Martingales and Brownian Motion,
Springer, New York, 1994.
H. Schurz: (A Brief Introduction
to) Numerical Analysis of (Ordinary) Stochastic Differential Equations
Without Tears, December Report 1670, IMA, Minneapolis, 1999
(Published by Marcel Dekker, Basel, 2002).
H. Schurz: Probabilidad de Honores, Universidad de Los Andes, Bogota,
1998, Lecture Notes Math 581 : Probability based on Measure Theory, SIU,
2002 (Original lecture script can be seen in my office).
A.N. Shiryaev: Probability, Springer, New York, 1996 (translation of
Russian original, Nauka, Moscow, 1980 (1989)).
C. Tudor: Procesos Estoc\'{a}sticos, Aportaciones Matem\'{a}ticas:
Textos 2, Sociedad Matem\'{a}tica Mexicana, M\'{e}xico City, 1994.
(565 pp., ISBN: 968-36-4004-4).
A.D. Wentzell: A Course in the Theory of Stochastic Processes,
McGraw-Hill, New York, 1981.