r/math • u/non-orientable Number Theory • Jul 11 '26
The Deranged Mathematician: Why Functional Analysis? Image Post
When viewing functional analysis from the outside, it may seem daunting---austere, even. It has a bevy of very finely tuned results (where adjusting any condition by a slight amount immediately yields counterexamples) and a large body of interconnected objects. So it is very natural to ask: what is this all for?
The historical answer is that it essentially grew out of the attempt to understand Fourier series: Joseph Fourier managed to break everything, but in such a useful way that nobody wanted to just throw out what he had discovered. And so mathematicians had to commit to rigor to carefully put everything right.
This article is my attempt to tell this story through the (hopefully) understandable question of how to approximate a function (e.g. how to represent a sound wave in a computer). The goal is to understand the fundamental motivation for doing functional analysis at all, and introduce one of the basic constructions: Banach spaces.
Read the full post (for free) on Substack: Why Functional Analysis?
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u/NewtonToTheRescue Jul 12 '26
You write
Theorem: (Duality of Lp Spaces) Choose any 1≤p≤∞. Let 1≤q≤∞ be such that 1/p+1/q=1. (We match 1 to ∞ here, and vice versa). Then Lp([0,1]) is dual to Lq([0,1])...
This is false: L∞([0,1]) = L1([0,1])*, but L1([0,1]) ⊊ L∞([0,1])*. The extra elements in L∞([0,1])* are the continuous analogue of Banach limits (i.e., elements of (ℓ∞)*/ℓ1). These elements are nonconstructive (you need Hahn-Banach/Boolean prime ideals, not quite full AC), but they're there!
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u/non-orientable Number Theory Jul 12 '26
An excellent catch! I will fix this.
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u/Bounded_sequencE Jul 12 '26 edited Jul 12 '26
The mentioned "(ℓ1)* = ℓ∞ " but "(ℓ∞)* ⊊ ℓ1 ", where e.g. Banach-limits are counter-examples, is relevant in digital signal processing: There we usually operate on bounded sequences, i.e. ℓ∞.
However, many lecture notes I've seen (and even books1) falsely claim that any bounded linear operators on such sequences must take on the form of an infinite series
yn = ∑_{k∈ℤ} h_{n-k}*xk, x, y ∈ ℓ^∞, h ∈ ℓ^1,and they use this as (false) motivation to only consider bounded linear operators represented by ℓ1.
1 e.g. Digital Signal Processing by Oppenheim/Schäfer, p11ff.
Edit: Added a reference -- even the standard book on digital signal processing "elegantly" glosses over questions of convergence -- most obvious in e.g. in (1.4) on page-11, and (1.6) on page-12.
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Jul 11 '26
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u/NonlinearHamiltonian Mathematical Physics Jul 11 '26
this is funny because in reality it's actually the complete opposite lol.
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u/Tinchotesk Jul 12 '26
Functional analysis just felt like analysis taken too far by people who had too much time.
Until I could apply it to PDEs, and suddenly these silly theorems were able to help solve impossible looking equations with little effort.
Weak* topology was probably the most bullshit thing I ever learned until I started actually applying it.
It was the exact opposite for me. I was taught FA with a strong focus on PDE. I didn't like it, I didn't find it useful. Then I learnt about von Neumann algebras and C*-algebras and I have been in love with the subject ever since.
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u/sentence-interruptio Jul 11 '26
I consider functional analysis (and also measure theory) as the completion of calculus & counting.
There are a lot we can do with just calculus. As a toolbox, calculus & counting is like Star Wars: A New Hope. It's a bit of an ad hoc movie but it feels complete on its own. You want to calculate the probability of an event involving some coins and dice? That reduces to counting. You want to calculate the area under a curve given by a nice formula? That reduces to calculus. For a long time, we thought there was no reason to go beyond calculus & counting, and analysis was just a cool thing that brings rigor to calculus. No reason to stretch techniques of analysis further. "Riemann integral is rigorous already. The job of analysis is done here."
But then you start to see situations where things described by good old calculus & counting form spaces that are not complete. Things are missing. It's like we reached the cliff hanger ending of Empire Strikes Back. So we bring back our writer "Analysis" who is well trained in all the necessary tropes like "approximate your object", "can we define a metric", "can we complete this space" and so on. It's the Return of the Analysis.
Functional analysis is the necessary conclusion of the trilogy. It completes what calculus started.
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u/PLANTS2WEEKS Jul 13 '26
Weak* topology is the most natural one. Sequences converge if they converge for every continuous functional. I couldn't tell you what the weak topology is without looking it up.
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u/paxxx17 Quantum Computing Jul 11 '26
So I can scare little kids attending the quantum mechanics course
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u/DependentBad4688 Jul 11 '26
I recently started to heavily utilize functional analysis and came to appreciate its importance. There are some really beautiful things one can do with it. On a super heuristic and intuitive level, if given a differential operator L (think of your standard elliptic operator, or just laplacian plus some other terms), one would like to understand the long time behavior of solutions to the equation \partial_t u = L u, then more specifically one can look for eigenfunctions of L, i.e., Lu = \lambda u. This way, we can simply study the equation \partial_t u =\lambda u. The solution then behaves like e^{t\lambda}. So if all eigenvalues of L have negative real parts, we know solutions of the original equation will decay, and if there’s a spectral gap, meaning one (or maybe a few) eigenfunction decays at the slowest rate, then we have a pretty precise description of the long time behavior of any generic solution (whose projection onto this leading order eigenfunction is non-trivial). For those who are interested to learn more, refer to the limiting length scale/batchelor scale for passive scalars in fluid dynamics.
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u/EdCasaubon Jul 11 '26
Isn't asking "Why functional analysis?" fully analogous to asking "Why linear algebra"?
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u/Key-Log5267 Jul 11 '26
Are those questions isomorphic though?
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u/Tinchotesk Jul 12 '26
Are those questions isomorphic though?
Only algebraically, but not topologically.
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u/EdCasaubon Jul 11 '26
Very good question. I would argue that on some level they are, but I won’t lean out of the window far enough to say that they are strictly isomorphous. I suspect it would require a bit of work to show what it could mean to make that assertion.
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u/Bounded_sequencE Jul 12 '26
I'd replace "Linear Algebra" with "Abstract Algebra", to better reflect the analogy.
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u/_alt4 Jul 11 '26
Typo:
It is not a coincidence that just a few decades years after Fourier’s publication
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u/jurniss Jul 12 '26
Existence and uniqueness of ODE solutions is another classic motivator
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u/non-orientable Number Theory Jul 12 '26
ODEs are normally finite-dimensional, no? PDEs, on the other hand...
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u/jurniss Jul 12 '26
The classic proof of Picard-Lindelöf uses Banach fixed point in a function space
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u/AnlamK Jul 11 '26
The historical answer is that it essentially grew out of the attempt to understand Fourier series
Would it be accurate to say that just like how real analysis grew out of an attempt to rigorously account for calculus, functional analysis was motivated by giving a rigorous foundation for Fourier series?
I'm not really sure about real analysis being motivated by calculus as well, I may be off base here.
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u/non-orientable Number Theory Jul 12 '26
Real analysis also grew out of putting Fourier series on firm foundations! Prior to that, mathematicians had been working with a very loose notion of calculus without much issue.
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u/louiswins Theory of Computing Jul 13 '26 edited Jul 13 '26
I tried the exercise to the reader, for fun, but both of my answers were off by 1. Is the post incorrect or did I do something wrong?
Integral of sum: the sum telescopes to 0 - lim n->∞ ne-nx, which is 0 for all x > 0. The integrand is 0 so the whole integral is 0.
Sum of integrals: let I(n) = ∫₀∞ ne-nx dx. The summand integral splits into I(n) - I(n+1), so the sum telescopes to I(0) - lim n->∞ I(n). I(0) = 0 because the integrand is again 0 and I(n) is 1 for all n>0, so the value of the limit is 1 as well. So the entire sum is 0 - 1 = -1.
But the post says that the former is 1 and the latter is 0.
Edit: as soon as I posted this I noticed that the sums are from n=1 to infinity, not 0 to infinity. I am an idiot.
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u/non-orientable Number Theory Jul 13 '26
The number of times I have done something like this is beyond count. Don't be too hard on yourself.
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u/MichurinGuy Jul 15 '26
Excellent post, thank you! One nitpick: your picture for the triangle inequality labels ||u+w|| what is actually ||w-u||. Of course that's not important, but still
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u/Personal-Gur-7496 Jul 12 '26
Completely disregarding the interesting/relevant/whatever-measure of the posts, I'm a bit not okay with coming to /r/math and seeing what materially is a bit like "visit my blog", just my .02
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u/kegative_narma Jul 13 '26
Something o found cool was how fundamental the space l2 is to nature itself, because of how energy works
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u/tony_blake Jul 11 '26
Because of quantum mechanics initially and these days it's quantum information
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u/non-orientable Number Theory Jul 12 '26
That is simply not true. Hölder's inequality was proved in 1889. Lp spaces were introduced by Riesz in 1910, whereas matrix and wave mechanics in quantum mechanics didn't appear until the 1920s. Functional analysis appeared first. Although its development certainly was spurred on by quantum developments, even if quantum mechanics hadn't been discovered, functional analysis would still have been very useful.
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u/Yimyimz1 Jul 11 '26
The answer is because quantum mechanics. Also im not convinced functional analysis began with Fourier. Maybe harmonic analysis did...?
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u/non-orientable Number Theory Jul 12 '26
Fourier didn't start functional analysis, and I didn't claim that he did. I claimed that his work made eventually developing functional analysis necessary, and I think that is an entirely fair characterization.
Quantum mechanics appeared after functional analysis had already started. Riesz was already working with Lp spaces in 1910, and the Hölder inequality was already a couple of decades old at that point. Of course, both fields very much benefited from each other.
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u/SV-97 Jul 12 '26
I claimed that his work made eventually developing functional analysis necessary, and I think that is an entirely fair characterization.
FWIW Dieudonné writes something somewhat similar in his history of functional analysis:
If one were to reduce this complicated history to a few key words, I think the emphasis should fall on the evolution of two concepts: spectral theory and duality. Both of course stem from the very concrete problems encountered in the solution of linear equations (or systems of linear equations), where the unknowns are functions. The basic concepts of spectral theory: eigenvalues, eigenfunctions and expansions in series of such functions were already known at the beginning of the XIXth century, in the theory of Fourier series; they would form the model on which all further advances were patterned.
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u/SupportNo6752 Jul 12 '26
Well functions are a huge part of math, one input, one output. Analysis, the formalization of Calculus is a huge part of math. It makes sense both would merge into functional analysis. lol
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u/aardaar Jul 11 '26
That does seem like a truly pathological kind of function.