*Review by Francesco Pedulla‘. Francesco is a Mechanical Engineer by education, an Information Technology Architect by profession, occasionally an (Adjunct) University Professor.*

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Functional Analysis is the foundation of two topics I have a deep interest for: Quantum Mechanics and the Finite Element Method. At my university, back in the ’80s, it was not part of the curriculum for Engineers. As I did with other topics of interest, I tried to learn more by myself and found it really hard to master. For sure, I could not find the right textbook for self-teaching. Or maybe nobody wrote it yet.

I regularly visit Coursera, that is my preferred source for MOOCs. When I saw a class with the inviting title of “An Introduction to Functional Analysis”, I had a look at the prerequisites and I thought: “Here it is! I found it at last – this is my opportunity to understand FA!”. Then, I had a look at the syllabus and I thought: “OMG! This is going to be really hard!”. Both thoughts happened to be right.

An Introduction to Functional Analysis by John Cagnol

## LECTURER

The main teacher is Professor John Gagnol. He looks younger than most professors I happened to meet at the University – with the remarkable exception of my Mathematical Analysis professors. Is it a chance? For sure, his role as Dean of Engineering qualifies him well for teaching FA to an engineer – or even a physicist, I guess. He presented all the lectures, while his assistant (Anna Rozanova-Pierrat) wrote the handout notes. His explanations are usually very clear, though sometimes a bit dry – typical of math classes. He speaks slowly, with a strong French accent – easy to follow anyway (but I am Italian, which probably helps).

## REQUIREMENTS

The Coursera web page for the class recommends “familiarity with functions, derivatives and integrals”. I totally agree and I add: the more, the better. The web page also states that you need “ to know what a proof is”. I suspect that’s not enough: you need to be able to build proofs, or at least be very motivated to learn how to. This is a class that will appeal to both scientists and engineers, people who are trained to understand what a proof “means” in physical terms rather than build their own proofs. If you are not a mathematician, beware: most assignments are essentially theorems, not the kind of exercises you find in most applied math classes for non-mathematicians. This is in line with the French mathematical tradition and I enjoyed it. But it can be overwhelming if you are not motivated enough.

## THE COURSE

The course lasts eight weeks. Each week you have to listen to about one hour of brief lectures, occasionally interspersed with quizzes, read the notes and do your homework (standalone quizzes and assignments). The course covers a lot of stuff, starting with very basic (but not trivial) concepts in topology and building up to the full glory of Hilbert and Sobolev spaces with application to a typical engineering problem (a membrane).

The handout notes are almost a book on FA: 167 pages, readable but terse. After adding the text of the assignments and their solutions, I printed out a book of about 200 pages, that now proudly sits on my bookshelf.

The assignments were six, plus the final, and consisted of proving a set of statements, usually quite general. For example, the first assignment was: *Let A ⊂ (X, T ). Prove that A = A iff A is closed*. For your info, *A* is a topological space built on the set *X* with topology *T*. Not as hard as it may sound – especially if you never studied topology before (like myself). The proof that I wrote was over two pages long (see figure) and took me several hours of work. The assignments are peer-graded, which means you have to grade the homeworks of (at least) five other students. As a consequence, your homework will be graded by five students. The grading process proved harder than I expected: first you need to understand very well the provided solution, which becomes the benchmark for the evaluation. Then, you have to carefully read someone else’s proof. In several cases, other students’ explanations were poor or too terse and errors could show up at any point. Some handwritten documents were even hard to read. But I noticed that the level of the proofs had a positive time derivative… :-). I assumed that many less prepared students were giving up as the class progressed. The final was really hard: I wrote down a 10 pages long document (roughly twice the reference solution, largely because I am a little less terse than the typical mathematician).

The web page of the class reports that the course should take you 6-8 hours per week. It may be true for students of mathematics or very mathematically inclined scientists. It took me on average well over 12 hours a week, all included. Much depends on the number of exercises from the handouts you try to do. I tried almost all of them, which proved essential to correctly approach many assignments.

## CONCLUSION

This is the hardest MOOC I ever attended. I learned a lot, even in areas (like topology) that I did not quite expect. But each and every argument the class touched upon had its role in building the comprehension of how – for example – even a non continuous function can be seen as the solution of a partial differential equation. If you are interested in this kind of stuff – and willing to learn how to write a mathematical proof – then this class is the best you can find.

I gave this class five stars, that are well deserved. I send a warm “thank you” to Prof. Gagnol for freely sharing his knowledge and deep understanding of such a complex subject.

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