Graduate Course 22M:313

Functional Analysis

Fall 2007

Some facts and observations:

I will aim at coordination between the two courses, and in fact you can take them in parallel. While 313 starts at the beginning, we will aim at some modern applications, including wavelets, fractals, and some engineering tools. Selection and choices depend on student demand!

See also (will be updated)

Course Meeting Schedule:

MWF 3:30 in MLH 205.

Course Books:

Book: Analysis and Probability: Wavelets, Signals, Fractals (Graduate Texts in Mathematics 234) (Hardcover) by Palle E.T. Jorgensen (Author).

Cheap Supplement: Elementary Functional Analysis (Paperback) by Georgi E. Shilov .

Introductory text covers basic structures of mathematical analysis (linear spaces, metric spaces, normed linear spaces, etc.), differential equations, orthogonal expansions, Fourier transforms.

There will be an organizational session in the first class meeting where we will discuss selection of topics, emphasis, and course requirements etc, but by tradition for 300 level courses, they are flexible and can be tailored to your plans.

PS.: 300 does not mean that it is more difficult!

I am also happy if you ask me when run into me, or you may stop by my office, 22B.

PS.: The supplement books (the two books by Georgi E. Shilov) are in the Dover math series. Inexpensive reprints of classics!

Relevant Seminars:

FYI: The two seminars, Operator Theory and Math Physics for fall 2006.

One organizational meeting Tuesday 8/22 at 1:30 in VAN 301 (that's the Physics building!)

Two slots: 1:30, and 2:30.

General course info:

Copies of the Syllabus are placed in the basket outside my office door. We will discuss HW + Exams, and class projects as the course progresses.

What is Functional analysis?

Function spaces, operators in Hilbert space, operator theory/operator algebras, mathematical physics (especially quantum theory), representations of groups, mathematical probability theory.

The answer (to the question "What is Functional analysis?") changes over time, and some answers refer to the mathematical tools, while others to a variety of applications; see e.g., the article by Vershik (handout!)

Topics and names from Vershik's article (P.S. Functional analysis is alive and well!): S. Banach's three theorems: (a) Hahn-Banach (extensions), (b) H. Steinhaus (uniform boundedness), and (c) the inverse mapping theorem (closed graph.)

1. Abstract vs concrete.

Felix Klein, J. Bernoulli (KAM), Cantor, Bourbaki, L. Schwartz (distributions.)

Additional references:

Schwartz, Laurent; Some applications of the theory of distributions. 1963 Lectures on Modern Mathematics, Vol. I pp. 23--58 Wiley, New York.

Kuksin, Sergei B.; Fifteen years of KAM for PDE. Geometry, topology, and mathematical physics, 237--258, Amer. Math. Soc. Transl. Ser. 2, 212, Amer. Math. Soc., Providence, RI, 2004.

Lax, Peter D. Jürgen Moser, 1928--1999. Ergodic Theory Dynam. Systems 22 (2002), no. 5, 1337--1342.

Raussen, Martin; Skau, Christian; Interview with Peter D. Lax. Reprinted from Eur. Math. Soc. Newslett., September 2005, 24--31. Notices Amer. Math. Soc. 53 (2006), no. 2, 223--229.


(a) Fourier series.
A. Zygmund, N. Wiener, A. N. Kolmorogov , L. Carleson.

(b) PDE:
Sobolev, P. Lax.

(c) Numerical analysis
(finite elements, G. Strang.)

(d) Calculus of variations
(Bernoulli's brachistocrone problem, Dirichlet and other boundary value problems.)

Additional references:

Rudin, Walter; Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1.

Nirenberg, Louis; Laurent Schwartz and some of his mathematical work. Laurent Schwartz (1915--2002). Gaz. Math. No. 98, suppl. (2003), 97--103.

Lax, Peter D.; Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. xx+580 pp. ISBN: 0-471-55604-1

3. Banach-algebras, C*- and W*-algebras. Special topics in algebras of Functional analysis:

(a) KMS

(b) Representations and states:
GNS (Gelfand-Naimark-Segal),

(c) Tomita-Takesaki.

(d) BDF
(Brown-Douglas-Fillmore); see 5 (c).

Additional references:

Bratteli, Ola; Robinson, Derek W.; Operator algebras and quantum statistical mechanics. Vol. 1. $C\sp{*} $- and $W\sp{*} $-algebras, algebras, symmetry groups, decomposition of states. Texts and Monographs in Physics. Springer-Verlag, New York-Heidelberg, 1979. xii+500 pp. ISBN: 0-387-09187-4.

4. Convexity:

Krein-Milman, G. Choquet, S. Kakutani. Applications: (a) optimization, (b) linear programming, (c) game theory, (d) infinite-dimensional analysis.

Additional references:

Rudin, Walter; Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. ISBN: 0-07-054236-8.

Phelps, Robert R.; Lectures on Choquet's theorem. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London 1966 v+130 pp.


(a) Hilbert space,

(b) Banach space:
J. von Neumann, Krein-Gohberg, Foias-Nagy, P. Enflo.

(c) Algebraic methods:
BDF (Brown-Douglas-Fillmore; famous 1977 Annals paper)

Additional references:

von Neumann, Johann; Mathematische Grundlagen der Quantenmechanik. (German) Unveränderter Nachdruck der ersten Auflage von 1932. Die Grundlehren der mathematischen Wissenschaften, Band 38 Springer-Verlag, Berlin-New York 1968 v+262 pp.

Sz.-Nagy, Béla; Foia\lfhook s, Ciprian; Harmonic analysis of operators on Hilbert space. Translated from the French and revised North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest 1970 xiii+389 pp.

6. Early quantum theory:

W. Heisenberg (uncertainty relations), E. Schrodinger, the Uniqueness theorem of Stone-von Neumann, quantum computing.

Additional references:
Shor, Peter W.; Introduction to quantum algorithms. Quantum computation: a grand mathematical challenge for the twenty-first century and the millennium (Washington, DC, 2000), 143--159, Proc. Sympos. Appl. Math., 58, Amer. Math. Soc., Providence, RI, 2002.

7. Statistical thermodynamics:

infinite-dimensional integration; Wiener, Kolmogorov, Feynman-Kac, Segal.

Additional references:

Nelson, Edward; Mathematical form and physical reality. (Italian) Stochastic processes in classical and quantum systems (Ascona, 1985), 545--550, Lecture Notes in Phys., 262, Springer, Berlin, 1986.

Nelson, Edward; Construction of quantum fields from Markoff fields. J. Functional Analysis 12 (1973), 97--112.

8. Representations of groups:

E. Wigner, V. Bargmann, G. Mackey.

Additional references:

Mackey, George W.; Mathematical foundations of quantum mechanics. With a foreword by A. S. Wightman. Reprint of the 1963 original. Dover Publications, Inc., Mineola, NY, 2004. xii+137 pp. ISBN: 0-486-43517-2.

9. Information theory, entropy, quantum computing:

Ehrenfest, A. Khinchin, R. Shannon.

Additional references:

de Leeuw, K.; Moore, E. F.; Shannon, C. E.; Shapiro, N.; Computability by probabilistic machines. Automata studies, pp. 183--212. Annals of mathematics studies, no. 34. Princeton University Press, Princeton, N. J., 1956.

Shannon, Claude E.; Weaver, Warren The Mathematical Theory of Communication. The University of Illinois Press, Urbana, Ill., 1949. vi+117 pp.

10. Wavelets:

the second course book (by Jorgensen), and

Additional references:

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, 1992.

O. Bratteli and P.E.T. Jorgensen, Wavelets through a Looking Glass: The World of the Spectrum, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2002.

11. Mathematical probability theory:

A. N. Kolmorogov (extensions revisited), random walk, information theory; Lebesgue-Rokhlin.

Additional references:

Nelson, Edward; Notes on non-commutative integration. J. Functional Analysis 15 (1974), 103--116.

12. Duality for groups:

Abelian vs nonabelian: L. S. Pontryagin, W. Rudin, V. Bargmann, G.W. Mackey, I.E. Segal.

Additional references:

Rudin, Walter; Fourier analysis on groups. Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1990. x+285 pp. ISBN: 0-471-52364-X.

Mackey, George W.; The scope and history of commutative and noncommutative harmonic analysis. History of Mathematics, 5. American Mathematical Society, Providence, RI; London Mathematical Society, London, 1992. xii+370 pp. ISBN: 0-8218-9000-X.

13. K-theory:

Atiyah-Singer, uses in C*- and W*-algebras. A. Connes, BDF (Brown-Douglas-Fillmore); see 5 (c), and J. Cuntz.

Additional references:

Brown, L. G.; Douglas, R. G.; Fillmore, P. A.; Extensions of C*-algebras and K-homology. Ann. of Math. (2) 105 (1977), no. 2, 265--324.

14. Symbolic dynamics:

Lind-Marcus: An Introduction to Symbolic Dynamics and Coding (Paperback) by Douglas Lind, Brian Marcus.

Additional references:

Shannon, Claude E.; Von Neumann's contributions to automata theory. Bull. Amer. Math. Soc. 64 (1958), 123--129.

Four Lists of Books Relevant to 22M:313 at

PS.: There are links to all of Jorgensen's book lists, reviews and other activities at this link:
Palle E.T. Jorgensen's amazon Profile.
Palle E. T. Jorgensen

This page was last modified on 29 June 2007 by Myung-Sin Song.

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