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.

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.

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.

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.