# What do I need to know in advance before taking a course in Functional Analysis?

Do I need a course on measure theory, or could I get by with just picking it up along the way, during the Functional Analysis course? Does the course just use some main results -- lebesgue measure + integration, and doesn't use the full course material of a measure theory course?

Anything else?

One more thing: Does the material in Functional Analysis and Abstract Algebra complement each other well?

I have a full year of graduate-level Linear Algebra completed. I also know non-measure-theoretic real analysis and complex analysis.

Thanks,

• It depends, but examples in functional analysis are often taken from the realm of Lebesgues integrable functions, so a solid knowledge of measure theoretic basics will be helpful. You should ask the instructor, if possible. May 12 '16 at 19:56
• Even if you haven't taken a course in Measure Theory, it is fairly easy to pick up if you are good with topology.
– user204299
May 12 '16 at 23:59

## 2 Answers

This is the description on math.stackexchange.com for Functional Analysis:

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modelled by these vector spaces.

Whoever wrote that tag description did a fairly decent job of describing Functional Analysis. There are some basic ingredients: Infinite-dimensional spaces, topology that allows approximation, continuous operators and continuous linear functionals. Lebesgue integration is nice in such settings, but not necessary. However, general operator theory on function spaces is not so easily studied without Measure Theory.

An introductory course may focus on metric topology, inner product spaces, and normed linear spaces. The classical integration spaces using Riemann integration do not give you complete spaces, but you can define an abstract completion where the original spaces are densely embedded. The completion will end up being a Lebesgue space, but you don't necessarily need to know that in order to work with the abstract completion. You can study continuous functionals, dual spaces, continuous linear operators, spectral theory on complete spaces, as well as operator algebras, without digging into the specifics of Lebesgue spaces, but the examples won't be as thoroughly investigated without Measure Theory. However, the general theory will be just as powerful and meaningful without such details, and classical ordinary differential operators can be effectively studied using only Riemann integration, including the existence of orthogonal bases of functions and integral transforms such as the Fourier transform.

A more advanced course will typically deal with topological vector spaces, which are vector spaces equipped with general topologies that give continuous vector and scalar multiplication operations. The study of more general spaces requires General Topology and convex analysis. Examples include spaces of distributions and other advanced topics in Partial Differential Equations. Sobolev spaces and embeddings can be effectively studied if you know Lebesgue integration. Advanced versions of functional calculus can be studied in this setting.

It might depend on who is teaching the class, but a little bit of knowledge of measure theory can be useful to put things in perspective, especially since a large number of examples and counterexamples involve $L^p$ and $\ell^p$ spaces.

What scares me is that there is no mention of Topology in your question. I'd say General Topology (what is covered in chapter 2,3,4,5,7,8 in Munkres' book, just to give a reference) is a prerequisite for Functional Analysis.

• Hi @Giovanni - so, the basic point-set topology, e.g., topology on $\mathbb{R}$ and topology on $\mathbb{R^n}$ from my non-measure-theoretic real analysis courses are not sufficient, you think? Thanks, May 12 '16 at 20:21
• @User001, I believe that good grasp of general (compact) Hausdorff spaces is necessary as the Banach-Alaoglu theorem lies at the heart of many functional analytic arguments. May 12 '16 at 20:26
• Compactness (or the lack of it) is usually a big deal in functional analysis, so, in my opinion, you might miss the big picture without a deeper knowledge of topology. As Thomas suggested, you can't go wrong if you ask the instructor :) May 12 '16 at 20:27
• Functional analysis classes/books often contain the topology that they use. Still, there is a lot of it, so being acquainted with it from the outside is probably a good idea if you have a chance.
– Ian
May 13 '16 at 0:15