Functional analysis math explained!

D. A. B. Miller, “An introduction to functional analysis for science and engineering” arXiv:1904.02539

The topic known as functional analysis is one that is sometimes taught to mathematics students, but typically falls just outside the math that scientists and engineers learn. Possibly as a result, the texts in this field, though they can be quite complete, are written in a heavily mathematical style that is particularly forbidding to others outside math itself. For some science and engineering fields – for example, in understanding waves – functional analysis can, however, deliver some very powerful results with major physical implications.

An example problem is understanding just how many channels we have for communicating with waves. We cannot prove a key result – that this number is always finite if we ask for finite coupling strenghts – without functional analysis. I wrote this tutorial on functional analysis as I was exactly trying to prove this and other results for waves (see D. A. B. Miller, “Waves, modes, communications, and optics: a tutorial,” Adv. Opt. Photon. 11, 679-825 (2019) https://doi.org/10.1364/AOP.11.000679 ).

In one sense, there is nothing new in this tutorial. However, I have written this in a style that I think is much more accessible to readers who are not necessarily mathematicians, while still including all the key proofs and results we need. Because I am more focused in getting to specific end points, this tutorial is also about a factor of 10 shorter than a typical math text in the field. Hopefully, then, this can serve a useful function for a broader audience, and it may also give math students an alternative (and possibly, even for them, more accessible!) introduction.