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.
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).
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.
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.
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.
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.
@UCLAExtension I know a follow up course to the first half of Differential Topology is being offered for Winter 2022, but it doesn’t seem to be on the site yet to register. Can someone fix this?
https://www.uclaextension.edu/sciences-math/math-statistics/course/introduction-differential-topology-math-x-45148
It hasn’t been announced officially in the UCLA Extension catalog, but Dr. Mike Miller’s anticipated course topic for Fall 2021 is differential topology. The anticipated recommended text is Differential Topology: An Introduction by David B. Gauld (M. Dekker, 1982 or Dover, 1996 (reprint)).
The offering is naturally dependent on potential public health measures in September, which may also create a class limit on the number of attendees, so be sure to register as soon as it’s announced. For those who are interested in mathematics, but have never attended any of Dr. Miller’s lectures, I’ve previously written some details about his stye of presentation, prerequisites (usually very minimal despite the advanced level of the topics), and other details.
A few of us have already planned weekly Thursday night topology study sessions through the end of Spring and into Summer for those interested in attending. Just leave a comment with your contact information and I’ll be in touch with details.
I hope to see everyone in the fall.
34.177879-118.101555
Last night saw the wrap up of Dr. Michael Miller’s excellent Winter quarter class Introduction to Category Theory. As usual he passed out a short survey to accept ideas for the Fall and Winter quarters this coming year at UCLA Extension.
If you didn’t get a chance to weigh in, feel free to email him directly, or respond here with your suggestions (in order of preference) and I’ll pass them along.
I keep a list of his past offerings (going back to 2006, but he’s been doing this since 1973) on my site for reference. He’s often willing to repeat courses that have been previously offered, particularly if there’s keen interest in those topics.
Some of the suggestions on last night’s list included:
combinatorics
combinatorial group theory
number theory
game theory
group theory
ring theory
field theory
Galois theory
real analysis
point set topology
differential equations
differential geometry
Feel free to vote for any of these or suggest your own topics. Keep in mind that many of the topics in the past decade have come about specifically because of lobbying on behalf of students.
This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra.
Category theory, since its development in the 1940s, has assumed an increasingly center-stage role in formalizing mathematics and providing tools to diverse scientific disciplines, most notably computer science. A category is fundamentally a family of mathematical obejcts (e.g., numbers, vector spaces, groups, topological spaces) along with “mappings” (so-called morphisms) between these objects that, in some defined sense, preserve structure. Taking it one step further, one can consider morphisms (so-called functors) between categories. This course is an introduction to the basic tenets of category theory, as formulated and illustrated through examples drawn from algebra, calculus, geometry, set theory, topology, number theory, and linear algebra. Topics to be discussed include: isomorphism; products and coproducts; dual categories; covariant, contravariant, and adjoint functors; abelian and additive categories; and the Yoneda Lemma. The course should appeal to devotees of mathematical reasoning, computer scientists, and those wishing to gain basic insights into a hot area of mathematics.
January 8, 2019 - March 19, 2019
Tuesday 7:00PM - 10:00PM
Location: UCLA
Instructor: Michael Miller
Fee: $453.00
The new catalog is out today and Mike Miller’s Winter class in Category Theory has been officially announced.
Oddly, it wasn’t listed in the published physical catalog, but it’s available online. I hope that those interested in mathematics will register as well as those who are interested in computer science.
Last night was the first lecture of Dr. Miller’s Gems And Astonishments of Mathematics: Past and Present class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.
Lecture notes
For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)
We’ve covered the following topics:
Class Introduction
Erdős Discrepancy Problem
n-cubes
Hilbert’s Cube Lemma (1892)
Schur (1916)
Van der Waerden (1927)
Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.
Course Description
Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.
Suggested Prerequisites
Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.
I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.
The newly renovated classroom space in UCLA’s Math Sciences Building
Category Theory for Winter 2019
As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!
As I get amped up for the start of Mike Miller’s Fall math class Gems and Astonishments of Mathematics, which is still open for registration, I’m even more excited that he’s emailed me to say that he’ll be teaching Category Theory for the Winter Quarter in 2019!!
Okay so right now I go to coffee shops to solve math problems alone, it's peaceful, I like it
But someone mentioned they do cute tea parties with their girl squad & I said wow I want something like that but we all bring math textbooks & solve problems next to each other (1/2)
It’s not specifically femme yet does involve tea, but I’ve noticed something informal like this at the Starbucks just two blocks West of CalTech in Pasadena.
Separately but related, “adults” looking for a varied advanced math outlet in the Los Angeles area are welcome to join Dr. Mike Miller’s classes at UCLA Extension on Tuesday nights from 7-10pm. We’re working on Algebraic Geometry this quarter. For those who might need notes to play catch up, I’ve got copies, with full audio recordings, that I’m happy to share.
Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.
Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.
Alright math nerds, it’s that time again! Be sure to register for Mike Miller’s excellent follow-on course for Algebraic Geometry.
Don’t forget to use the coupon code EARLY to save 10% with an early registration–time is limited!
Stephanie Hurlburt