Today’s reading and writing space.
Category: Mathematics
This textbook has evolved from a set of lecture notes which I prepared for a semester course in Hilbert space. I have in mind first- or second-year graduate students in Mathematics and related fields such as Physics.
My textbook for Dr. Miller’s class Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions has arrived.
Dr. Miller has tipped me off that the suggested text for his upcoming course An Introduction to Hilbert Spaces will be:
Berberian, Sterling Khazag. Introduction To Hilbert Space. Oxford University Press, 1961. Reprint Literary Licensing, 2012.
He’s not happy that it ignores measure theory as a means to introduce the Lebesque integral, so he’ll be supplementing that with additional notes. I’ve ordered a used copy of the 1st edition, but there are also versions from AMS as well as a more recent reprint from 2012.
He also suggested that Debnath & Mikusinski was pretty good, albeit more expensive than he would like in addition to not being a fan some of their approaches to topics.
Debnath, Lokenath, and Piotr Mikusinski. Introduction to Hilbert Spaces with Applications. 3rd ed., Academic Press, 2005.
Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions
Looking for some serious entertainment with an intellectual bent on Tuesday nights this fall? Professor Michael Miller has got you covered in multiple dimensions.
Dr. Miller has now listed his mathematics offering for Fall 2025 at UCLA Extension. It’s Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions (MATH 900). As always, it will be presented in lectures on Tuesday nights from 7:00 PM to 10:00 PM with a short break in the middle. The class runs from September 23 – December 9 and is a screaming deal at just $450.00.
As many know, Dr. Miller does a superb job presenting advanced and abstract mathematics to the point that most students who take one or two classes return for decades. If you’re a fan of math and physics and have wanted to delve beneath the surface, this is an excellent opportunity to not only begin, but to meet lots of others who share your interests. For newcomers interested in taking a peek, I’ve written up a short introduction to his teaching style with some hints and tips based on my 18 years of taking coursework with him in his 52 year teaching career. There’s definitely a reason dozens of us keep showing up.
Here’s the description in the course catalog:
This course is designed for scientists, engineers, mathematics teachers, and devotees of mathematical reasoning who wish to gain a better understanding of a critical mathematical discipline with applications to fields as diverse as quantum physics and psychology.
A Hilbert space is a vector space that is endowed with an inner product for which the corresponding metric is complete (i.e., every Cauchy sequence converges). Examples include finite-dimensional Euclidean spaces; the space l2 of all infinite sequences (a1, a2, a3, …) of complex numbers, the sum of whose squared moduli converges; and the space L2 of all square-summable functions on an interval. This introductory, yet rigorous, treatment focuses initially on the structure (orthogonality, orthonormal bases, linear operators, Bessel’s inequality, etc.) of general Hilbert spaces, with the latter part of the course devoted to interpreting these constructs in the context of Legendre polynomials, Fourier series, Sobolev spaces, and other prominent mathematical structures.
The listed prerequisites for the course are calculus and linear algebra, though Dr. Miller generally does an excellent job of bringing up students without a huge machinery of mathematics background or sophistication up to speed to appreciate the material. Whatever you do, don’t let the technical nature of the description deter you from jumping into abstract mathematics with both feet.
Bibliography
The UCLA Bookstore currently doesn’t have a suggested textbook for the course listed. Dr. Miller doesn’t require a textbook, but will often suggest one in addition to the incredibly comprehensive notes he provides in his lectures for understanding the subject. For the curious and the less-experienced or budding mathematicians out there, his lecture notes are clearer and imminently more understandable than any book you’re likely to find on the subject.
For those curious in exploring the space, I’ve put together a short bibliography of some of the more common textbooks covering the undergraduate and graduate studies within the area. Dr. Miller is sure to choose one at the level of an advanced undergraduate (junior or senior level).
- Akhiezer, N. I., & Glazman, I. M. (1993). Theory of Linear Operators in Hilbert Space. Dover Publications.
- Alabiso, C., & Weiss, I. (2021). A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups. Springer.
- Debnath, L., & Mikusinski, P. (2005). Introduction to Hilbert Spaces with Applications. Academic Press.
- Fuhrmann, P. A. (2014). Linear Systems and Operators in Hilbert Space. Dover Publications.
- Halmos, P. R. (1982). A Hilbert Space Problem Book. Springer.
- Halmos, P. R. (2017). Introduction to Hilbert Space and the Theory of Spectral Multiplicity (2nd ed.). Dover Publications.
- Kolmogorov, A. N., & Fomin, S. V. (1961). Elements of the Theory of Functions and Functional Analysis, Volume 2, Measure. The Lebesgue Integral, Hilbert Space. Greylock Press.
- Muscat, J. (2024). Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras. Springer.
- Nagy, B. S., Foias, C., Bercovici, H., & Kérchy, L. (2010). Harmonic Analysis of Operators on Hilbert Space. Springer.
- Stein, E. M., & Shakarchi, R. (2005). Real Analysis: Measure Theory, Integration and Hilbert Spaces. Princeton University Press.
- Sunder, V. S. (2016). Operators on Hilbert Space. Hindustan Book Agency.
- Szekeres, P. (2006). A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry. Cambridge University Press.
- Weidemann, J., & Szücs, J. (1980). Linear Operators in Hilbert Spaces. Springer. (Original work published 1976)
- Young, N. (1997). An Introduction to Hilbert Space. Cambridge University Press.
Ever thought about studying mathematics for fun?!?
As a reminder for local Los Angeles area hobbyist mathematicians and physicists, Dr. Miller will be teaching Fundamentals of Point-Set Topology at UCLA starting September 24th. I hope some new folks will join our merry band for some math fun this fall. First time taking some math after high school/college? I’ve got some tips here.
Point-set topology is the branch of mathematics that deals with collections of points endowed with sufficient structure to make meaningful the notions of closeness, separation, and convergence. Beginning with familiar notions concerning open sets, closed sets, and convergence on the real number line and Euclidean plane, this course systematically develops the theory of arbitrary topological spaces. Topics include bases and subbases, separation axioms (Hausdorff, regular, and normal spaces), countability (first- and second-countable spaces), compactness and compactification, connectedness, and convergence (nets and filters). Instruction emphasizes examples and problem solving. The course appeals to those seeking a better understanding of the algebraic and geometric underpinnings of common mathematical constructs.
September 24 - December 3 on Tuesday 7:00PM - 10:00PM PT
Fee: $453.00
Location: UCLA, Math Sciences Building, Room 5127
Mike Miller’s fall math class at UCLA has been posted. I’m registering and hope to see you there!
As usual, there’s no recommended textbook (yet), and he generally provides his own excellent notes over a required textbook. I’d suspect that he’ll recommend an inexpensive Dover Publication text like those of Kahn, Baum, or Gamelin & Greene.
If you’re curious about what’s out there, I’ve already compiled a bibliography of the usual suspects in the space:
- Armstrong, M. A. Basic Topology. Undergraduate Texts in Mathematics, 3.0. Springer, 1983.
- Conover, Robert A. A First Course in Topology: An Introduction to Mathematical Thinking. Reprint. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2014.
- Conway, John B. A Course in Point Set Topology. Undergraduate Texts in Mathematics. Springer, 2015.
- Crossley, Martin D. Essential Topology. Corrected printing. Springer Undergraduate Mathematics Series. 2005. Reprint, Springer, 2010.
- Gaal, Steven A. Point Set Topology. 1st ed. Pure & Applied Mathematics 16. Academic Press, 1964.
- Gamelin, Theodore W., and Robert Everist Greene. Introduction to Topology. 2nd ed. Dover Books on Mathematics. 1983. Reprint, Mineola, N.Y: Dover Publications, Inc., 1999.
- Kahn, Donald W. Topology: An Introduction to the Point-Set and Algebraic Areas. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 1995.
- Kasriel, Robert H. Undergraduate Topology. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2009.
- López, Rafael. Point-Set Topology: A Working Textbook. 1st ed. Springer Undergraduate Mathematics Series. Springer, 2024.
- Mendelson, Bert. Introduction to Topology. 3rd ed. Dover Books on Mathematics. Dover Publications, Inc., 1990.
- Morris, Sidney A. Topology Without Tears, 2024. [.pdf]
- Munkres, James R., 1930-. Topology. 2nd ed. 1975. Reprint, Prentice-Hall, Inc., 1999.
- Shick, Paul L. Topology: Point-Set and Geometric. 1st ed. Wiley-Interscience, 2007.
- Sierpinski, Waclaw. General Topology. Translated by C. Cecilia Krieger. Repring. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2020.
- Viru, O. Ya., O.A. Ivanov, N. Yu. Netsvetaev, and V.M. Kharlamov. Elementary Topology: Problem Textbook. American Mathematical Society, 2008.
- Waldmann, Stefan. Topology: An Introduction. Springer, 2014.
- Willard, Stephen. General Topology. Dover Books on Mathematics. Mineola, N.Y: Dover Publications, Inc., 2004.
AI generated featured photo courtesy of Glif Alpha
Dr. Mike Miller announced in class last night that in the coming Winter quarter at UCLA Extension he’ll be offering a course on elliptic curves.
The text for the class will be Rational Points on Elliptic Curves (Springer, Undergraduate Texts in Mathematics) by Joseph H. Silverman and John T. Tate. He expects to follow and rely more on it versus handing out his own specific lecture notes.
He mentioned that while it would suggest a more geometric flavor, which it will certainly have, the class will carry an interesting algebraic component which those not familiar with the topic may not expect.
To register, look for the listing sometime in the coming month or so when the Winter catalog is released.
Topics to be discussed include the isomorphism theorems; ascending and descending chain conditions; prime and maximal ideals; free, simple, and semi-simple modules; the Jacobson radical; and the Wedderburn-Artin Theorem.
Ring theory is a branch of abstract algebra that deals with sets—for example, the collection of all integers—that admit both additive and multiplicative operations. Modules generalize the notion of vector spaces, but with scalars drawn from a ring rather than a field. Beginning with a survey of the basic notions of rings and ideals, the course explores some of the elegant algebraic structuring that defines the behavior of rings—both commutative and non-commutative—and their associated modules. Topics to be discussed include the isomorphism theorems; ascending and descending chain conditions; prime and maximal ideals; free, simple, and semi-simple modules; the Jacobson radical; and the Wedderburn-Artin Theorem. Theory will be motivated by numerous examples drawn from familiar realms of number theory, linear algebra, and real analysis.
Mike Miller has announced this fall’s math course at UCLA.
Quantum mechanics anyone? Dozens have been disappointed by UCLA’s administration ineptly standing in the way of Dr. Mike Miller being able to offer his perennial Winter UCLA math class (Ring Theory this quarter), so a few friends and I are putting our informal math and physics group back together.
We’re mounting a study group on quantum mechanics based on Peter Woit‘s Introduction to Quantum Mechanics course from 2022. We’ll be using his textbook Quantum Theory, Groups and Representations:An Introduction (free, downloadable .pdf) and his lectures from YouTube.
Shortly, we’ll arrange a schedule and some zoom video calls to discuss the material. If you’d like to join us, send me your email or leave a comment so we can arrange meetings (likely via Zoom or similar video conferencing).
Our goal is to be informal, have some fun, but learn something along the way. The suggested mathematical background is some multi-variable calculus and linear algebra. Many of us already have some background in Lie groups, algebras, and representation theory and can hopefully provide some help for those who are interested in expanding their math and physics backgrounds.
Everyone is welcome!
Are you a math person looking for an interesting space on Mastodon? Try https://mathstodon.xyz/about
Dr. Miller mentioned in class tonight that his Winter quarter math class at UCLA will be on ring theory. Prepare yourself to commute (or not).
Theory and Applications of Continued Fractions MATH X 451.50 | Fall 2022
For the Fall 2022 offering Dr. Michael Miller is offering a mathematics course on Theory and Applications of Continued Fractions at UCLA on Tuesday nights through December 6th. We started the first class last night, but there have been issues with the course listing on UCLA Extension, so I thought I’d post here for any who may have missed it. (If you have issues registering, which some have, call the Extension office to register via phone.)
For almost 300 years, continued fractions—that is, numbers representable as the sum of an integer and a fraction whose denominator is itself such a sum—have fascinated mathematicians with both their remarkable properties and their myriad applications in such fields as number theory, differential equations, and computer algorithms. They have been applied to piano tuning, baseball batting averages, rational tangles, paper folding, and plant growth … the list goes on. This course is a rigorous introduction to the theory and mathematical applications of continued fractions. Topics to be discussed include quadratic irrationals, approximation of real numbers, Liouville’s Theorem, linear recurrence relations and Pell’s equation, Hurwitz’ Theorem, measure theory, and Ramanujan identities.
Mike is recommending the Continued Fractions text by Aleksandr Yakovlevich Khinchin. I found a downloadable digital copy of the 1964 edition (which should be ostensibly the same as the current Dover edition and all the other English editions) at the Internet Archive at Based on my notes, it looks like he’s following the Khinchin presentation fairly closely so far.
If you’re interested, do join us on Tuesday nights this fall. (We’ve already discovered that going 11 for 37 is the smallest number of at bats that will produce a 0.297 batting average.)
If you’re considering it and are completely new, I’ve previously written up some pointers on how Dr. Miller’s classes proceed: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension
I just couldn’t wait for a physical copy of The First Astronomers: How Indigenous Elders Read the Stars by Duane Hamacher, Ghillar Michael Anderson, Ron Day, Segar Passi, Alo Tapim, David Bosun and John Barsa (Allen & Unwin, 2022) to arrive in the US, so I immediately downloaded a copy of the e-book version.
@AllenAndUnwin @AboriginalAstro
What is a "fundamental" unit of meaning or conceptual atom?
— Codex OS (@codexeditor) February 25, 2022
It is not commensurate with brevity. If we take as an atom "freedom" or "zero" we find these units are actually more complex.
Specify, then. But how to be so without involving hundreds of units?@alanlaidlaw @Conaw
@brunowinck @codexeditor @alanlaidlaw When thinking about this, recall that in the second paragraph of The Mathematical Theory of Communication (University of Illinois Press, 1949), Claude Shannon explicitly separates the semantic meaning from the engineering problem of communication.
