Functions Are Vectors (2023)
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>If you’re alarmed by the fact that the set of all real functions does not form a HILBERT SPACE, you’re probably not in the target audience of this post."
Video: https://youtu.be/q8gng_2gn70?t=8m3s
Thanks to
> Dr. von Neumann, ich möchte gerne wissen, was ist denn eigentlich ein Hilbertscher Raum? (Dr. von Neumann, I'd would really like to know, just what exactly is a Hilbert space?)
Asked to John von Neumann to David Hilbert at a lecture.
https://ncatlab.org/nlab/show/Hilbert+space#fn:1
I'd like to add, as a physicist by training, that anything can be a Hilbert space if you wish hard enough. You can even use results about countable vector spaces if you need them!
Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problemsAnd he's starting from the assumption vectors are finite (cf. the article)
You may recall that this representation is only one example of an abstract vector space. There are many other types of vectors, such as lists of complex numbers, graph cycles, and even magic squares.
However, all of these vector spaces have one thing in common: a finite number of dimensions. That is, each kind of vector can be represented as a collection of N N numbers, though the definition of “number” varies.
When you only have a finite number of functions (and the space they span), then you can apply your finite dimensional linear algebra, since n dimensional vector spaces over a field are isomorphic. You absolutely can gain intuition for real functions from arrows on a paper type vectors.
E.g. the article mentions the Cauchy-Schwartz inequalities for functions, that's something you can intuit when you imagine our functions being little arrows on a paper.
Basically it's an introduction to "functional analysis", the field of math that looks at functions like vectors. The term is in there sneakily, but this is really what the article is all about.
I had classes on this subject and we were encouraged to build upon linear algebra intuition (much like the author does), with some caveats (such as a bounded closed set is not necessarily compact in infinite dimension). Things break down, as the author hints at when they mention they didn't prove the Laplacian to be self-adjoint but only symmetric (this has to do with the operator's domain, a concern that doesn't exist for linear operators on finite dimensional spaces)... but it's still a very convenient way to think and talk about things, conceptually.
The observation here is that set of real value functions, combined with the set of real numbers, and the natural notion of function addition and multiplication by a real number satisfies the definition of a vector space. As a result all the results of linear algebra can be applied to real valued functions.
It is true that any vector space is isomorphic to a vector space whose vectors are functions. Linear algebra does make a lot of usage of that result, but it is different from what the article is discussing.
we’ve built a vector space of functions
Ideally, we could express an arbitrary function f as a linear combination of these basis functions. However, there are uncountably many of them—and we can’t simply write down a sum over the reals. Still, considering their linear combination is illustrative:
This vector space also has a basis (even if it is not as useful): there is a (uncountably infinite) subset of real->real functions such that every function can be expressed as a linear combination of a finite number of these basis functions, in exactly one way.
There isn't a clean way to write down this basis, though, as you need to use Zorn's lemma or equivalent to construct it.
I think what I may be asking is “Does the complex Fourier transform make a Hilbert space?” but I might be wrong both about that and about that being the right question.
You can represent any function f: [-pi, pi] -> R as an infinite sum
f(x) = sum_(k = 0 to infinity) (a_k sin(kx) + b_k cos(kx))
This is very useful, but the functions sin(x), sin(2x), ... , cos(x), cos(2x), ... don't constitute a basis in the formal sense I mentioned above as you need an infinite sum to represent most functions. It is still often called a basis though.
f(x) = sum n=-infty to +infty C_n e^{i n x}
sin x = (e^(ix) - e^(-ix))/2i
cos x = (e^(ix) + e^(-ix))/2
The convergence criteria for Fourier series vary depending on how strongly you need convergence but I think basically if a function is differentiable on the interval you care about then the Fourier series provably converges on that interval and otherwise if it has jump discontinuities and that sort of thing, then depending on whether it is square-integrable or a bunch of other properties) you can prove weaker forms of convergence (absolute, pointwise etc).
To address your comment I don’t see why an infinite sum prevents something being a basis. In fact I would specifically say that can’t be true because then there would never be a basis for any infinite-dimensional space- any time you want to take an inner product in such a space you need an infinite sum, and you need such an inner product to construct the basis. A sibling comment pointed me in the direction of a Hilbert basis, which seems to be what I was thinking of.
Another example is the eigenvectors of linear operators like the Laplacian. Recall how, in finite dimension, the eigenvectors of a full rank operator (matrix) form an orthonormal basis of the vector space. There is a similar notion in infinite dimension. I can't find an English page that covers this very well, but there's a couple of paragraphs in the Spectral Theorem page (https://en.wikipedia.org/wiki/Spectral_theorem#Unbounded_sel... ). The article linked here also touches on this.
Regarding your last sentence, one thing to note is that having a basis is not what makes you a Hilbert space, but rather having an inner product! In fact, to get the Fourier coefficients, you need to use that inner product.
However, the particular vector space in question (functions from R to R) does have a basis, which the author describes. That basis is not as useful as a basis typically is for finite dimensional (or even countably unfitine dimensional) vector spaces, but it still exists.
(Imagine/remember what it feels like when children first learn that the integers aren't the only number, there are also fractions, then irrationals, then complex numbers...this is a very similar situation).
With that in mind, you may want to reread the text and pay attention to the definitions he is using, and not assume that your definitions are the whole story.
(Imagine/remember what it feels like when children first learn that the integers aren't the only number, there are also fractions, then irrationals, then complex numbers...this is a very similar situation).
> However, the particular vector space in question (functions from R to R) does have a basis, which the author describes.
No, there is no known constructible basis for R -> R functions.
This is _also_ true. But the fact that finite dimensional real vectors can be viewed as a special case of set-theoretic functions or as a special case of real functions on a discrete finite space is probably less useful than the opposite: the set of real functions have a vector-space structure, and you can use all the neat theorems about (finite or infinite dimensional) vector spaces.
Well, at least the article is about the latter direction.
- How much of this structure survives if you work on "fuzzy" real numbers? Can you make it work? Where I don't necessarily mean "fuzzy" in the specific technical sense, but in any sense in which a number is defined only up to a margin of error/length scale, which in my mind is similar to "finitism", or "automatic differentiation" in ML, or a "UV cutoff" in physics. I imagine the exact definition will determine how much vectorial structure survives. The obvious answer is that it works like a regular Fourier transform but with a low-pass filter applied, but I imagine this might not be the only answer.
- Then if this is possible, can you carry it across the analogy in the other direction? What would be the equivalent of "fuzzy vectors"?
- If it isn't possible, what similar construction on the fuzzy numbers would get you to the obvious endpoint of a "fourier analysis with a low pass filter pre-applied?"
- The argument arrives at fourier analysis by considering an orthonormal diagonalization of the Laplacian. In linear algebra, SVD applies more generally than diagonalizations—is there an "SVD" for functions?
Perhaps some conjugate relation could be established between finite-range in one domain and finite-resolution in another, in terms of the effect such nonlinearities have on the spectral response.
2/3. I'm not really sure what you mean by these questions... But if you want to do "fourier analysis with a filter preapplied", you'd probably just work with within some space of bandlimited functions. If you only care around N Fourier modes, any time you do an operation which exceeds that number of modes, you need to chop the result back to down to size.
4. In this context, it's really the SVD of an operator you're interested in. In that regard, you can consider trying to extend the various definitions of the SVD to your operator, provided that you carefully think about all spaces involved. I assume at least one "operator SVD" exists and has been studied extensively... For instance, I can imagine trying to extend the variational definition of the SVD... and the algorithms for computing the SVD probably make good sense in a function space, too...
As a result we get finite resolution and truncation of the spectrum. So "Fourier analysis with pre-applied lowpass filter" would be analysis of sampled signals, the filter determined by the sampling kernel (delta approximator) and properties of the DFT.
But so long as the sampling kernel is good (that is the actual terminology), we can form f exactly as the limit of these fuzzy interpolations.
The term "resolution of the identity" is associated with the fact that delta doesn't exist in most function spaces and instead has to be approximated. A good sampling kernel "resolves" the missing (convolutional) identity. I like thinking of the term also in the sense that these operators behave like the identity if it were only good up to some resolution.
If you wanted something more quantized, you can pick some length unit, d, and replace the real numbers with {... -2d, -d, 0, d, 2d,... }. This forms a structure known as a "ring" with the standard notion of addition, subtraction, and multiplication (but no notion of division. Using this instead of R does lose the vector structure, but is still an example of a slightly more general notion of a "module". Many of the linear algebra results for vector spaces apply to modules as well.
> If it isn't possible, what similar construction on the fuzzy numbers would get you to the obvious endpoint of a "fourier analysis with a low pass filter pre-applied?"
If that is where you want to end up, you could pretty much start there. If you take all real value functions and apply a courier analysis with a low pass filter to each of them, the resulting set still forms a vector space. Although I don't see any particular way of arriving at this vector space by manipulating functions pre Fourier transform.
[1] https://proceedings.neurips.cc/paper_files/paper/1999/file/9...
The popular lens is the porcupine concept when infinite dimensions for functions is often more effective when thought of as around 8:00 in this video.
While that video obviously is not fancy, it will help with building an intuition about fixed points.
Explaining how the dimensions are points needed to describe a functions in a plane and not as much about orthogonal dimensions.
Specifically with fixed points and non-expansive mappings.
Hopefully this helps someone build intuitions.
I guess it works if you look at it sideways.
To me, the proper way of continuing to develop intuition is to abandon visualization entirely and start thinking about the math in a linguistic mode. Thus, continuous functions (perhaps on the closed interval [0,1] for example) are vectors precisely because this space of functions meet the criteria for a vector space:
* (+) vector addition where adding two continuous functions on a domain yields another continuous function on that domain
* (.) scalar multiplication where multiplying a continuous function by a real number yields another continuous function with the same domain
* (0) the existence of the zero vector which is simply the function that maps its entire domain of [0,1] to 0 (and we can easily verify that this function is continuous)
We can further verify the other properties of this vector space which are:
* associativity of vector addition
* commutativity of vector addition
* identity element for vector addition (just the zero vector)
* additive inverse elements (just multiply f by -1 to get -f)
* compatibility of scalar multiplication with field multiplication (i.e a(bf) = (ab)f, where a and b are real numbers and f is a function)
* identity element for scalar multiplication (just the number 1)
* distributivity of scalar multiplication over vector addition (so a(f + g) = af + ag)
* distributivity of scalar multiplication over scalar addition (so (a + b)f = af + bf)
So in other words, instead of trying to visualize an infinite-dimensional space, we’re just doing high school algebra with which we should already be familiar. We’re just manipulating symbols on paper and seeing how far the rules take us. This approach can take us much further when we continue on to the ideas of normed vector spaces (abstracting the idea of length), sequences of vectors (a sequence of functions), and Banach spaces (giving us convergence and the existence of limits of sequences of functions).
My third way is that I learn math by learning to "talk" in the concepts, which is I think much more common in physics than pure mathematics (and I gravitated to physics because I loved math but can't stand learning it the way math classes wanted me to). For example, thinking of functions as vectors went kinda like this:
* first I learned about vectors in physics and multivariable calculus, where they were arrows in space
* at some point in a differential equations class (while calculating inner products of orthogonal hermite polynomials, iirc) I realized that integrals were like giant dot products of infinite-dimensional vectors, and I was annoyed that nobody had just told me that because I would have gotten it instantly.
* then I had to repair my understanding of the word "vector" (and grumble about the people who had overloaded it). I began to think of vectors as the N=3 case and functions as the N=infinity case of the same concept. Around this time I also learned quantum mechanics where thinking about a list of binary values as a vector ( |000> + |001> + |010> + etc, for example) was common, which made this easier. It also helped that in mechanics we created larger vectors out of tuples of smaller ones: spatial vector always has N=3 dimensions, a pair of spatial vectors is a single 2N = 6-dimensional vector (albeit with different properties under transformations), and that is much easier to think about than a single vector in R^6. It was also easy to compare it to programming, where there was little difference between an array with 3 elements, an array with 100 elements, and a function that computed a value on every positive integer on request.
* once this is the case, the Fourier transform, Laplace transform, etc are trivial consequences of the model. Give me a basis of orthogonal functions and of course I'll write a function in that basis, no problem, no proofs necessary. I'm vaguely aware there are analytic limitations on when it works but they seem like failures of the formalism, not failures of the technique (as evidenced by how most of them fall away when you switch to doing everything on distributions).
* eventually I learned some differential geometry and Lie theory and learned that addition is actually a pretty weird concept; in most geometries you can't "add" vectors that are far apart; only things that are locally linear can be added. So I had to repair my intuition again: a vector is a local linearization of something that might be macroscopically, and the linearity is what makes it possible to add and scalar-multiply it. And also that there is functionally no difference between composing vectors with addition or multiplication, they're just notations.
At no point in this were the axioms of vector spaces (or normed vector spaces, Banach spaces, etc) useful at all for understanding. I still find them completely unhelpful and would love to read books on higher mathematics that omit all of the axiomatizations in favor of intuition. Unfortunately the more advanced the mathematics, the more formalized the texts on it get, which makes me very sad. It seems very clear that there are two (or more) distinct ways of thinking that are at odds here; the mathematical tradition heavily favors one (especially since Bourbaki, in my impression) and physics is where everyone who can't stand it ends up.
If you told me this in the first year of my math degree I would have included myself in that group. I think you’re right that a lot of people are filtered out by higher math’s focus on definitions and theorems, although I think there’s an argument to be made that many people filter themselves out before really giving themselves the chance to learn it. It took me another year or two to begin to get comfortable working that way. Then at some point it started to click.
I think it’s similar to learning to program. When I’m trying to write a proof, I think of the definitions and theorems as my standard library. I look at the conclusion of the theorem to prove as the result I need to obtain and then think about how to build it using my library.
So for me it’s a linguistic approach but not a natural language one. It’s like a programming language and the proofs are programs. Believe it or not, this isn’t a hand-wavey concept either, it’s a rigorous one [1].
[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
fwiw, this is exactly the thing that you when you're trying to formally prove some theorem in a language like Lean.
Right?! In my path through the physics curriculum, this whole area was presented in one of two ways. It went straight from "You don't need to worry about the details of this yet, so we'll just present a few conclusions that you will take on faith for now" to "You've already deeply and thoroughly learned the details of this, so we trust that you can trivially extend it to new problems." More time in the math department would have been awfully useful, but somehow that was never suggested by the prerequisites or advisors.
But when I did go past the required courses and into math for math majors, things got a lot better. I just didn't find that out until I was about to graduate.
Except none of this is true of vectors in general, although it might be true of very specific vector spaces in physics that you may have looked at. Matrices or continuous functions form vector spaces where you can add any vectors, no matter how far apart. Maybe what you're referring to is that differentiability allows us to locally approximate nonlinear problems with linear methods but that doesn't mean that other things aren't globally linear.
I also don't understand what you mean by "no difference between composing vectors with addition or multiplication", there's obviously a difference between adding and multiplying functions, for example (and vector spaces in which you can also multiply are another interesting structure called an algebra).
That's the problem if you just go from intuition to intuition without caring about the formalism. You may end up with the wrong understanding.
Intuition is good when guided by rigour. Terence Tao has written about this: https://terrytao.wordpress.com/career-advice/theres-more-to-...
The vector space axioms in the end are nothing more than saying: here's a set of objects that you can add and scale and here's a set of rules that makes sure these operations behave like they're supposed to.
The general theme is that I am interested in the metaphysical concept of vectors, not the thing that human mathematicians have labeled vectors. The universe doesn't care if you write ax+by or x^a y^b, hence addition vs multiplication is just a choice of coordinate system. And matrices and functions are vector spaces sure, but out in the world, when they show up in modeling things, they are local linearizations of curved things. Every linear algebra is (inevitably) a local point in a nonlinear one, as far as I can tell. Not in a formal sense, but in the sense that when you go out into the world and find them, it turns out to be the case.
The general theme is: I don't want to spend my life mastering the rigor of these simplistic models so that I can do it intuitively (in Tao's sense); I want to use them to learn intuition of the things that they are simplistic models of, and then master that.
That's fine but then you shouldn't be surprised that you can't read higher level mathematics textbooks, because those are not about metaphysics.
The rest of what you wrote is... just not true. Matrices are used in plenty of areas where they are not mere approximations (e.g. cryptography), and spaces of functions remain vector spaces even when the functions themselves are not linear (because there's a difference between requiring that f(x+y)=f(x)+f(y) and that (f+g)(x) = f(x)+g(x)).
Isn't this how people arrived at most of these concepts historically, how the intuition arose that these are meaningful concepts at all?
For example, the notion of a continuous function arose from a desire to explicitly classify functions whose graph "looks smooth and unbroken". People started with the visual representation, and then started to build a formalism that explains it. Once they found a formalism that was satisfying for regular cases, they could now apply it to cases where the visual intuition fails, such as functions on infinite-dimensional spaces. But the concept of a continuous function remains tied to the visual idea, fundamentally that's where it comes from.
Similalrly with vectors, you have to first develop an intuition of the visual representation of what vector operations mean in a simple to understand vector space like Newtonian two-dimensional or three-dimensional space. Only after you build this clean and visual intuition can you really start understanding the formalization of vectors, and then start extending the same concepts to spaces that are much harder or impossible to visualize. But that doesn't mean that vector addition is an arbitrary operation labeled + - vector addition is a meaningful concept for spatial vectors, one that you can formally extend to other operations if they follow certain rules while retaining many properties of the two-dimensional case.
This notion falls down when you get to topology where you have continuous functions on topological spaces (which need not have any concept of distance nor even "smoothness"), since a topology can be defined on a finite (or infinite) set (of objects which may not even be numbers).
But that doesn't mean that vector addition is an arbitrary operation labeled + - vector addition is a meaningful concept for spatial vectors, one that you can formally extend to other operations if they follow certain rules while retaining many properties of the two-dimensional case
Vector addition need not even look like addition. For example, the positive real numbers can be defined as a vector space over the real numbers, with:
* (+) vector addition: u + v = uv (adding two vectors by multiplying two positive real numbers)
* (.) scalar multiplication: av = v^a
* (0) zero vector: 1 (the identity for multiplication)
Now we can verify some of the vector space axioms:
* let a be a scalar and u, v be vectors, then: a(u + v) = a(uv) = (uv)^a = (u^a)(v^a) = au + av, thus distributivity of scalar multiplication over vector addition holds
* let a, b be scalars and v be a vector, then: (a + b)v = v^(a + b) = (v^a)(v^b) = av + bv, thus distributivity of scalar multiplication over scalar addition holds
The rest can also be similarly verified but you get the picture.
Another weird vector space is the set of spanning subgraphs of a finite, simple, undirected graph over the finite field F[2] (which yields only 0 and 1 as scalars). In this one the idea of vector addition between subgraphs G + H is about determining whether two vertices are adjacent in one or the other of G or H, or adjacent in neither or both. This isn't really like addition at all, so none of the intuitions you might develop for vector addition in a Euclidean two-dimensional space would apply at all.
What I claim in addition is that it's still useful for your intuition to understand this history and the leaps that were made by extremely talented mathematicians of the past who came up with these formalizations of intuitive properties.
I'd also claim that they couldn't have ever arrived at the current formal systems if they hadn't started with certain intuitions for simple systems.
What the person you replied to is suggesting, however, is that this can only get you so far. At some you have to sit down and study the formal definition and understand how it relates to the examples you've seen so far. Otherwise it's very easy to have your intuition lead you astray.
(This is where I learned at least half of the math on this page: theoretical chemistry.)
Let's see: Let A be a non-empty set, N the set of positive whole numbers, and X the set of all functions f
f: A --> N
with usual notation.
Assume as is common, the scalars are the set of real numbers, but the set of complex numbers will also do.
So, is X a vector space and, thus, each f in X a vector?
No, since -f is not in X. Neither is (1/2)f.
Some references (with TeX markup):
Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition\/}
linear algebra treated as functional analysis.
Walter Rudin, {\it Real and Complex Analysis\/}
with Lebesgue integration and, then, Banach and Hilbert vector spaces.
Walter Rudin, {\it Functional Analysis\/}
with Fourier theory.
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/}
with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.
This is what the article is very explicitly about. I guess you can quibble that the title is imprecise, but it's just a title and the article makes it clear.
Well, not with the operations pulled from the codomain at least.
What I did was follow, as in the references, long established convention, that for a function to be a vector at least it had to be in a vector space where (1) can multiply a function by a number (e.g., reals or complex) and (2) add two functions and still get a function in the vector space. To be general, I omitted metrics, inner products, topologies, convergence, probability spaces, and more.
Or, as in the references I gave, math talks about vector spaces and vectors, and each vector is in a vector space. The references are awash in definitions of vector spaces with (1) and (2) and much more.
Computing is awash in indexes for data, e.g., B-trees, SQL (structured query language) operations on relational data bases, addressing in central processors, collection classes in Microsoft's .NET, REDIS, and calling all such also functions confuses established material, conventions, and understanding.
An infinite sequence approximates a general function, as described in the article (see the slider bar example). In signal processing applications, functions can be considered (or forced) to be bandlimited so a much lower-order representation (i.e. vector) suffices:
- The subspace of bandlimited functions is much smaller than the full L^2 space - It has a countable orthonormal basis (e.g., shifted sinc functions) - The function can be written as (with sinc functions):
x(t) = \sum_{n=-\infty}^{\infty} f(nT) \cdot \text{sinc}\left( \frac{t - nT}{T} \right)
- This is analogous to expressing a vector in a finite-dimensional subspace using a basis (e.g. sinc)
Discrete-time signal processing is useful for comp-sci applications like audio, SDR, trading data, etc.
Basically, define a finite dimensional function space V_N of dimension N, in such a way that you could grow N to be arbitrarily large. Solve not the PDE (originally defined over an infinite dimensional function space V, such as H^1), but its discretization as if it dealt only with functions in V_N rather than all functions. The PDE is then simply a linear system, easy to solve. And you can prove, for instance in the case of elliptic PDEs, that the solution to the discrete problem is the orthogonal projection of the true solution of the PDE (in V) onto V_N (Céa's Lemma). Finally, you can produce estimations of the error this projection incurs as a function of N, and thus give theoretical guarantees that the algorithm converges to the true solution as N goes to infinity.
(N in this case is the number of vertices in a mesh that is used to define the basis functions of V_N)
Given an vector space V with (+, ), you can define the vector space over functions F whose codomain is V and where F.+ and F. both take two functions as argument and return another function applying V.+ or V.* on the result. All the linear algebra properties come from the original vector space. Hence it is boring.
That's absolutely not the case.
Can you elaborate? Which vector space(s) do these vectors belong to, as drawn?
If anything, this just highlights the sloppiness of these types of illustrations. Things aren't precise enough, in my opinion, for the illustrations to do anything except confuse.
In physics, a vector is often more specifically something with magnitude and direction. This still doesn't mean that it needs to be anchored at the origin. Vectors that are anchored at the origin are IIRC called position vectors, but mathematically, if you translate them away from the origin they're still the same vector.
The graph of a function f: X -> Y is the set {(x, f(x)) | x in X}. It is much more clear and precise to associate elements of the graph with vectors such that the 0 vector is identified with the R^2 origin, and then points in R^2 are identified with vectors. Then there is a mapping between vectors in this vector space to the graph, i.e., to points (x, f(x)).
> In physics, a vector is often more specifically something with magnitude and direction.
Physics is sloppy. :) This is not a general description of a vector, where vector is an element of a general vector space. Not all vector spaces have a norm, which is required for magnitude to make any sense.
> but mathematically, if you translate them away from the origin they're still the same vector.
Right, and you cannot always translate vectors without more machinery, such as parallel transport.
Fine, then let's be precise. An element of the vector space R^n is nothing more than a function from {1,...,n} to R. And every n-dimensional R-vector space is isomorphic to it.
How you wanna draw this in a coordinate system is up to you. It is customary to identify vectors with points on that coordinate system and then equivalently with an arrow pointing from the origin to that point. In that case the zero vector is the origin, or an arrow from the origin to itself.
It is equally customary to use vectors as displacement, i.e. a directed difference between two points. In that case, the vector that you can't actually draw because it has zero length is your zero element. Now your arrows don't have to be anchored at the origin anymore.
Both of these spaces are of course isomorphic.
If your vector space is infinite-dimensional it's of course not gonna be really possible to draw anymore.
When drawing vectors, such as on a 2D space, the origin is identified with this 0 vector, and then the points in the 2D space can be associated with the vectors in the vector space.
It's not a particularly interesting proof, but the author does prove that real valued functions are vectors. The bulk of the article is less about proofs, and more about showing how the above result is useful.
how the above result is useful
Here's the definition of a vector space which agrees with the one everyone in mathematics use: https://thenumb.at/Functions-are-Vectors/#vector-spaces
From this it's fairly easy to prove (and done in the article) that the set of all functions R->R is a vector space.
Polynomials come to mind.
You don't need to go into intricacies like metrics and approximations because that's more like analysis.
You can however talk about infinite dimensional vectors spaces and talk about projections onto finite subspaces.
Functions on a countable domain are sequences.
Vector spaces can have infinite dimension, so the "only" in the first sentence does not belong there.
The second sentence is also odd. How do you define "sequence"? Are there no finite sequences?
For the second sentence, he's right, we could also write (wrongly) an article titled "Functions are Sequences" and (try to) apply what we know about dealing with countable sequences to functions
Those practical uses are someone else's problem to solve (even if they rely on math to solve them), and they can write their own web pages on how functions as vectors help solve specific problems in a way that's more insightful than using "traditional" calculus, and get those upvoted on HN.
But this link has a "you must be this math to ride" gate, it's not for everyone, and that's fine. It's a world wide web, there's room for all levels of information. You need to already appreciate the problems that you encountered in non-trivial calculus to appreciate this interpretation of what a function even is and how to exploit the new power that gives you.
My suggestion is that briefly mentioning them up front might be nice. I didn't mean to start a big argument about it.
Some people would like to have a filter for what to spend their time on, better than "your elders before you have deemed these ideas deeply important". One such filter is "Can these ideas tell us nontrivial things about other areas of math?" That is, "Do they have applications?"
Short of the strawman of immediate economic value, I don't think it's wrong to view a subject with light skepticism if it seemingly ventures off into its own ivory tower without relating back to anything else. A few well-designed examples can defuse this skepticism.
It's fun to simulate one thing with another, but there is a deeper and more profound sense in which vectors are functions in Clifford Algebra, or Geometric Algebra. In that system, vectors (and bi-vectors...k-vectors) are themselves meaningful operators on other k-vectors. Even better, the entire system generalizes to n-dimensions, and decribes complex numbers, 2-d vectors, quaternions, and more, essentially for free. (Interestingly, the primary operation in GA is "reflection", the same operation you get in quantum computing with the Hadamard gate)