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.