Part II of (Muscat 2024) looks at Banach and Hilbert Spaces.
Chapters 7, 8, 10 and 11 respectively cover normed spaces, bounded operators, Hilbert and Banach spaces. These chapters are the “heart” of the subject, defining its two basic spaces and the fundamental properties of operators. These topics appear in most courses on the topic. Chapter 11 was the most challenging for me, as some of the proofs were hard to unravel. For example, the Open Mapping Theorem took repeated attempts and comparisons with alternative presentations before I felt that I “got it”.
Chapter 9 steps away from the theory to present the key sequence and function spaces in which much of the work in this area occurs. Section 9.2 also provides a short introduction to measure theory - this is the first place where my specific background knowledge was lacking.
While skimming (Garcia, Mashreghi, and Ross 2023), I’ve become increasingly aware of the importance of complex analysis within functional analysis and beyond. Accordingly, Muscat provides some review and development of the subject, first in Chapter 12, which looks at differentiation and integration, and then in Chapter 13 of Part III. This was the second place where I needed to “top-up” my background knowledge - effectively, I went through the equivalent of an undergraduate course in the subject before returning to Muscat’s book.
Resources
Books:
- Measure, Integral and Probability (Capiński and Kopp 2004) - I read chapters 2, 3, 4 and 5, but only the sections dealing with the core theory, so skipping all the probability and mathematical finance parts.
- Complex Analysis (Howie 2003) - I read the whole book. This is an important subject, and also needed in Part III.
Exercises:
- For further practice, I am working through the problem book (Brayman et al. 2024) - its first 7 chapters cover the topics in Part II quite closely.
Videos:
- Complex Analysis - by Dr. Z. Sjostrom Dyrefelt, ICTP Diploma Programme. A complete course, at a slightly higher level than (Howie 2003) (e.g. presents theorems with winding numbers from the start).
- Functional Analysis - by Dr. Joel Feinstein, University of Nottingham - a complete course, from Metric Spaces to Banach Algebras, but with less detail than the book, so a useful introduction.
- Weak Convergence (three lectures) - by Dr. Melanie Rupflin, University of Oxford - taught as part of the third-year undergraduate course B4.2 Functional Analysis 2. These lectures complement section 11.5 and are worth watching.
I also found the following video courses useful:
- Complex Analysis - by Dr. Richard Borcherds, Fields Medallist. Covers many of the basics in an entertaining manner, and also discusses several interesting functions.
- Introduction to Functional Analysis - by Dr. Casey Rodriguez, MIT OpenCourseWare. More challenging, goes deeper into some topics.
- Introduction to Functional Analysis - by Prof. Lassi Paunonen, with notes. A lighter, high-level presentation.