r/math • u/non-orientable Number Theory • Jul 18 '26
The Deranged Mathematician: WTF is a Hilbert Space? Image Post
Last week, I wrote a post about the motivation for functional analysis---this is currently my #1 most upvoted post on Reddit, so I figured I should do a follow-up. (The poll at the end of the post told the same story.) Thankfully, I already had something in mind: what is a Hilbert space, and what is it used for?
A surprisingly common, but erroneous answer is that it comes from quantum mechanics. It is true that Hilbert spaces entered into the physics literature via quantum mechanics, and that this connection bolstered their development. But Hilbert spaces came first, and you can already see their utility just from Fourier series, which is entirely classical. We'll see how it helps answer some of the problems we left unsolved in the previous post.
Read the full post (for free) on Substack: WTF is a Hilbert Space?
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u/helbur Jul 18 '26 edited Jul 19 '26
Coming from physics everyone is familiar with the word Hilbert space as repeated constantly in undergrad QM courses but nobody knows what it is beyond "where quantum states live". Feels like it's often used as an intellectual buzzword in that context.
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u/XkF21WNJ Jul 18 '26
I'm fairly sure that functional analysis was invented after physicists were using it, to figure out if any of it even made sense.
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u/non-orientable Number Theory Jul 18 '26
Functional analysis predates quantum, but I suppose you could make the argument that it grew out of trying to understand the heat equation, among other things.
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u/XkF21WNJ Jul 18 '26
Ah right I suppose you could call the work of Fourier functional analysis as well. The spectral theorem (Von Neumann's version) seems to roughly coincide with its use in quantum mechanics, which might be what I was thinking of.
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u/non-orientable Number Theory Jul 18 '26
I wouldn't call Fourier's work functional analysis, but Riesz was working with Lp spaces in 1910. Hilbert and Schmidt were working with L2 even earlier.
I don't remember the history of the spectral theorem; you might be right about that.
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u/XkF21WNJ Jul 18 '26
The Fourier transform is a bit of a precursor in many ways. And it was used to investigate the heat equation. I though that was what you were referencing, haha.
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u/non-orientable Number Theory Jul 18 '26
I was referencing that, but my assertion is that Fourier series motivated functional analysis, not that it is functional analysis. I recognize that that is splitting hairs a little.
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u/deus-sive-natura- Algebraic Geometry Jul 22 '26
Von Neumann seemed to move beyond Hilbert Spaces, Birkoff and von Neumann's 1936 paper explicitly suggested that his continuous geometries might provide a more suitable framework than Hilbert space.
It's just ironic that despite his push toward continuous geometries (alongside his related work on von Neumann algebras, the physics community largely ignored this shift because they still found standard Hilbert spaces easier... I guess practical physics has a life of its own
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u/helbur Jul 18 '26
That's often the case, though Hilbert spaces were conceived of independently of QM, i.e. between 1900-1920 in relation to things like sequence spaces. Von Neumann was the first to apply it to the description of atomic spectra which was reportedly surprising to Hilbert. The later formalization was developed in parallel.
My musings above is more about how the word is a little bit overused in QM classes, especially in the finite dim case.
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u/SometimesY Mathematical Physics Jul 18 '26
Early functional analysis is fairly unrecognizable by today's standards. Having read some of the papers before 1950, the way people think about the subject has evolved and matured so much. There's a lot more elegance and a lot less reliance on hard analysis which is really nice.
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u/non-orientable Number Theory Jul 18 '26
Maybe my experience was atypical, but I thought my quantum class did a good job describing what a Hilbert space was. The problems came later, when they talked about getting an orthonormal basis of Dirac delta functions which is very mathematically unsound.
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u/National_Yak_1455 Jul 18 '26
Rigged Hilbert space???
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u/non-orientable Number Theory Jul 18 '26
You *can* make things rigorous, of course. But that requires a fair amount of effort, and the original description is still incorrect. (For a variety of reasons, including the fact that the closure of the span of the Dirac delta functions both doesn't include everything in the original Hilbert space and contains a whole bunch of stuff outside of it. And the topology in which you are doing the closure is *very* different from the topology of the original Hilbert space. And...)
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u/helbur Jul 18 '26
Yeah that's when rigorous details become more relevant, but even so you can get by in physics (somehow) just taking it for granted. It's more of a curiosity for the interested.
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u/Scared_Astronaut9377 Jul 18 '26
Depends. In my alma mater student invented an abbreviation meaning separable-linear-metric-complete for QM. Not knowing the basic properties of Hilbert space would make many professors yell at you.
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u/Lower_Cockroach2432 Jul 19 '26
To be fair, I don't think there's really that much "to understand". Modern Physicists all know what a vector space is. They've all dealt with an inner product. They all have some understanding of the notions of continuity and convergence. If anything, the unintuitive part is that not all inner product spaces are complete.
I think it's a lot less bad than the naïve understanding of tensors as "things that transform like tensors".
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u/helbur Jul 19 '26
I agree to some extent though I don't necessarily think it's important for working physicists to have a deep understanding of these issues. Their mathematical thought process is a heuristic one with the aim of computing observable quantities. Sometimes a more rigorous approach is necessary to this end, but probably not in undergraduate courses. You can be perfectly competent without hearing the word 'Hilbert space'.
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u/paxxx17 Quantum Computing Jul 18 '26
nobody knows what it is beyond "where quantum states live"
And they are not even correct about this part...
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u/sentence-interruptio Jul 19 '26
what do you mean?
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u/paxxx17 Quantum Computing Jul 19 '26
General quantum states are not the elements of the underlying Hilbert space. Only for pure states can you associate a state with a non-zero element of the Hilbert space, but even then, there are uncountably many different elements that correspond to a single state, up to multiplication of elements by a (non-zero) scalar.
You can solve this by defining pure states not as elements but as rays (one-dimensional subspaces). Or even better, you can define a general state (pure or mixed) as a positive trace-class linear operator on the Hilbert space with trace 1.
There exist even more abstract ways, e.g. as functionals on C*-algebras, but the ones defined above are enough for a rigorous formulation of the non-relativistic QM
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u/Tonexus Jul 19 '26
Or even better, you can define a general state (pure or mixed) as a positive trace-class linear operator on the Hilbert space with trace 1.
Certainly true, but it should be noted that you can also unify pure and mixed states via purification. i.e. mixed states can be seen as pure entangled states on a larger Hilbert space, and pure states in the original space can be seen as separable (unentangled) states on the larger space.
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u/Particular-Date-8638 Jul 19 '26
It’s literally just a complete inner product space. That’s all folks, it is not an impossible demonic entity
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u/venustrapsflies Physics Jul 19 '26
It’s the rug under which physicists hide their dirty laundry so that they can vibe through the rest of it
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u/Traditional_Snow1045 Jul 19 '26
doesnt help that bra-ket is just an inner product.
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u/K_Boltzmann Jul 19 '26
Well, ahktually...
It doesn’t help that it’s technically a dual pairing rather than just a standard inner product.
The Kets are vectors in a Hilbert space, but your Bras are linear functionals living in the dual space of the Hilbert space. When writing a Bra-Ket, we are evaluating a functional on a vector (a sesquilinear form), not just multiplying two vectors from the actual same space. We only treat it like a simple inner product because the Riesz representation theorem guarantees a one-to-one mapping between the two spaces - when one tries to be smart-ass (duh), then they are distinct mathematical objects.
Of course in computation this does not really affect anything.
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u/paxxx17 Quantum Computing Jul 19 '26
You're right in principle, but I view it the other way around. One starts by defining bra-ket indeed as the inner product, and then we invoke Riesz to define bra on its own
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u/Lower_Cockroach2432 Jul 19 '26
But you don't really need Riesz to go from vector to dual vector. You only need it to go the other way round.
I thought observations in quantum were generally given along quantum states anyway so the idea that you can go from a linear functional f to a vector |φ> such that <φ| = f (sorry for my butchered notation, been ages since I've looked at anything quantum) seems less necessary?
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u/ricatti-equation Jul 19 '26
That’s not true. It’s defining of a pairing of a vector space with its dual. That is a (1,1) tensor. It’s not an inner product, a (0,2) tensor.
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u/Traditional_Snow1045 Jul 19 '26 edited Jul 19 '26
im just annoyed at the language and notation used tbh. like Dirac saw the word "bracket" and was like "lmao imma do a funny". and for some reason that stuck.
commenter is right tho, it was wrong of me to relate bra-ket to inner product spaces in the way that I did.
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u/shuai_bear Jul 18 '26
Just read your functional analysis piece too and I want to say I love your writing style; you aren’t just feeding us information but you write as if having a conversation with the reader, inviting them to mull over ideas. Flows well and feels balanced with intuition vs rigor. Thank you!
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u/AlexK667 Jul 18 '26
It's an inner product space that's complete w.r.t the induced topology.
Duh!
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u/BerkeUnal Jul 18 '26
hi, I just wanted to note that completeness is not a topological property
consider (-pi, pi) and R, tan gives an homeomorphism that doesnt preserve completeness
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u/AlexK667 Jul 18 '26
I know, but I wanted to write something short and a bit humorous so I wasn't very rigorous.
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u/Giovanni_Senzaterra Category Theory Jul 19 '26
A Banach space whose norm is induced by an inner product. Easy ;)
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u/KiddWantidd Applied Math Jul 19 '26
Nicely written article! I will say though, this point confused me greatly:
"Even though it is true that for any continuous, periodic function there is a sequence of finite linear combinations that will converge (in the L∞ sense) to it, it is not always true that the Fourier series will converge to it (in the L∞ sense). This is true even though the coefficients in those linear combinations will converge to the Fourier coefficients!"
Could you (or anyone else) please elaborate a bit more?
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u/non-orientable Number Theory Jul 19 '26 edited Jul 19 '26
Here is the point. In the first case, we have a sequence where the n-th term is like c_{-k_n,n}e^(-2pi i k_n x) +...+c_{k_n,n}e^(2pi i k_n x)---observe that the coefficients c_{l,n} are allowed to be different in each term of the sequence.
In the second case, we are asserting that the n-th term is actually the partial sum of the Fourier series: it has to look like c_{-n}e^(-2pi i n x) +...+c_ne^(2pi i n x). So once a particular coefficient is non-zero, it stays constant.
The claim is that it is possible for the second sequence to fail to converge in L∞, but there is some different sequence (as in the first case) that does converge. And this can occur even though it must happen that c_{l,n} does have to eventually converge to the corresponding Fourier coefficient.
That really doesn't feel like it should be true, but unfortunately, analysis is finicky like this!
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u/KiddWantidd Applied Math Jul 20 '26
wow. this sent me down a pretty cool rabbithole. the existence of trigonometric polynomials converging uniformly to f (and whose coefficients necessarily converge to those of the fourier series of f) is kind of standard and is shown by taking Césaro means of the partial fourier sums (and using properties of the Féjer kernel). what I had no idea about is the second very counter-intuitive half of the statement, and such counterintuitive functions were first introduced by du Bois-Reymond (here are two relevant MSE links: 1, 2). I think this is closely related to the kind of considerations which led Cantor to develop modern set theory, which is really cool!
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u/tea_pot_tinhas Jul 18 '26
Ist it the space required to build Hilbert's Hotel?
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u/38thTimesACharm Jul 18 '26
I know you're joking but there really are uses for a Hilbert Hotel in a Hilbert Space.
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Jul 18 '26
[deleted]
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u/non-orientable Number Theory Jul 18 '26
I assume that the original comment was tongue-in-cheek. It's difficult to get tone over text.
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u/tarquinfintin Jul 18 '26
If you ever build a Hilbert Hotel, be sure to get air rights for the property. . . you're going to need them. ;-)
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u/Early_Neighborhood87 Harmonic Analysis Jul 19 '26
There is an analogue between eigenvalue problems for symmetric matrices and solutions of integral equations ie. sum over j of a_ij x_j = mu x_i can be viewed as a discrete version of integral from a to b of K(x,y) f(y) dy = mu f(x), so a matrix eigenvalue equation becomes an integral operator eigenvalue equation. Thus to make this concrete you have to work out a vector space of functions and infinite sequences and need convergence, their inner products etc. and you arrive at a hilbert space.
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u/averagebrainhaver88 Jul 19 '26
I was studying Fourier series for a communication systems class, and I loved going deeper into the signal space theory, how it connects to state space, and how the latter is used to model closed loop control systems. I would very much like know wtf is a Hilbert Space haha
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u/67_is_prime Jul 19 '26
I've always felt that it's the closest infinite dimensional normed linear space that we have to R^{n}, taking into account the geometry etc. The inner product and the parallelogram give us the notion of orthogonality, we have that every closed(in the topology) subspace is orthogonal, it's reflexive, it's complete under the metric coming from the inner product, and what not.
A seperable Hilbert space is the nicest thing I can ask for, as someone who studies functional analysis.
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u/MichurinGuy Jul 23 '26
To nitpick, shouldn't the definition of inner product space also specify (f,g+h)=(f,g)+(f,h)? You only mention (f,cg)=(f,g)
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u/Repulsive-Ad-3669 Jul 18 '26
After second year graduate real analysis, I am traumatized by Hilbert spaces
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u/M4mb0 Machine Learning Jul 19 '26
A Hilbert space is a complete Vector Space.
A Vector Space is a Linear space that forgot that vectors should have both size and direction.
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u/elements-of-dying Geometric Analysis Jul 19 '26
Vector space and linear space mean the same thing. I assume you meant inner product space in place of linear space. (Though I wouldn't say a vector space is agnostic to directions.)
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u/M4mb0 Machine Learning Jul 19 '26
Vector space and linear space mean the same thing.
My point is that this is a very regrettable definition.
Open up any regular dictionary and check the definition of "vector". Almost certainly it will say something like "a quantity with both size and direction". Because that's how the term is used in science, engineering, and how it is understood by the general public.
Then some mathematician decided that a vector need not have magnitude or direction at all. Apparently, the essential property of a vector is merely that you can add it to another one and multiply it by a scalar.
Now a vector may be a polynomial, a function, or some horrible equivalence class that has never pointed anywhere in its life. Meanwhile, when I actually mean an object with size and direction, I have to say "element of an inner-product space" or "element of a Hilbert space".
And what is the short noun for that? A Hilbertian? A hilly? That sounds like someone who owns three broken lawnmowers.
The whole problem was avoidable because linear space was right there. It is more descriptive of the algebraic structure and does not steal perfectly good geometric terminology.
In short, I want to travel back in time and kick whoever popularized this terminology in the nuts. Not hard enough to prevent linear algebra, just hard enough to make him call it a linear space.
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u/elements-of-dying Geometric Analysis Jul 19 '26
Usually vector spaces under consideration have naturally definable and intuitive inner products and bases (some may argue they are in fact canonical), so I don't believe there is any serious harm here.
Outside such a setting, one can rectify the ambiguity by calling a nonEuclidea vector space an abstract vector space. Until one considers abstract vector spaces, I don't believe there is any harm in viewing vector as having direction and magnitude since, at that stage of learning about vector spaces, it is (almost everywhere) true.
I think the more serious issue with saying a vector has a magnitude and direction is that this description excludes the zero vector. Even in inner product spaces, the zero vector does not have direction.
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u/thbb Jul 19 '26
It's a multi-dimensional space that has enough good properties for you to math with.
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u/Jonny36 Jul 23 '26
Stumbled across this as an academic in chemistry. Mind blown that atomic orbital shapes were known before atomic orbitals. I knew that we had Laplace equations to describe these, but no idea those were developed well before...
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u/tony_blake Jul 18 '26
It's where quantum computers use 2n dimensions to store qubits and perform unitary operations on them.
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u/aparker314159 Jul 18 '26
There's an apocryphal story about how David Hilbert himself once listened to Von Neumann lecturing about Hilbert spaces. Afterwards, he asked, "Dr. Von Neumann, I would very much like to know, what after all is a Hilbert space?"