We do it all the time. An index is just indicative that there is a mapping (a function), usually from the integers. However we don't use the subscript notation when indexing by a continuum due to the discomfort you describe.

The point is that we need some way to deal with objects that are inherently infinite-dimensional.

I'm probably ignorant of how indexes work at a nuts-and-bolts level, but intuitively this seems like a good idea for certain situations. E.g if we want to keep entries in a specific order but don't know ahead of time how many entries will be added between two existing ones. House numbers in areas with a lot of development are an example of the kind of problem this seems ideal to solve, when there's a clear 'order' based on geography but no clear limit on the number of addresses that could be added 'between' existing addresses.

The author is stretching an analogy, it's a price to pay for starting with R^3 as a motivational example. There is nothing in the general definition of a vector space that requires it's elements to be "indexed"

I think that’s why the author put “vector” in quotes. I kind of imagine it as an ephemeral, infinite list where for some real, when we use that real value as an index into our “vector”/function, we get the output value as the item in this infinite, ephemeral list.

I think the only thing that matters is that the indices have an ordering (which the reals obviously do) and they aren’t irrational (i.e. they have a finite precision).

Imagine you have a real number, say, e.g. 2.4. What stops us from using that as an index into an infinite, infinitely resizable list? 2.4^2 = 5.76. Depending on how fine-grained your application requires you could say 2.41 (=5.8081) is the next index OR 2.5 (=6.25) is the next index we look at or care about.

I could be misunderstanding it, though.

The only difference of note, I think, is that you can't enumerate the elements. Instead of being able to say "for each element, ..." you'd have to say "for all elements, ...", like the example of vector length defined as an integral over the full number range.

What do you understand “index” to mean here? To me, a family indexed by some set is mostly just a different notation for, and attitude towards, a function with domain the indexing set.

Consider a function on R as an |R|-dimensional vector...

You can look at use cases for an index, and see how well they hold up.

Asking where the smallest greater number (next number) is no longer makes sense.

Taking two numbers and asking whether one is greater than the other still makes sense. (and hence also whether they are equal)

Taking two numbers and asking how far separated from each other still makes sense.

You may already observe some uses for indexes in programming that don't use all of these properties of an index. For example, the index of a hash set "only cares about equality", and "the next index" may be an unfilled address in a hash set.

This Has it's use. The continouus Fourier Transform is is based on that. You are asking what frequencies is this continouus signal made of. Time is normally defined as a real number in that context, but If you have a continouus time you need continouus frequencies to map time space to frequency space. You can think about an Index as a lego Block, that you need to construct Something.

I think getting hung up on words (in this case index) in mathematics is a trap. They are often stretched to their breaking point and you just kind of go with the flow.

> When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’

> ’The question is,’ said Alice, ‘whether you can make words mean so many different things.’

> ’The question is,’ said Humpty Dumpty, ‘which is to be master — that’s all.

In mathematics it is the author's privilege.

Well there's no law against it.

Okay I suppose the axiom of choice is somewhat necessary to make it make sense. But only because otherwise such an indexed object may fail to exist.

Anyway arbitrary indexes are useful, you often end up doing stuff like covering a space by finding a covering set for each individual point. And then using compactness to show you only need finitely many to cover the whole space. It is doable without uncountable indices, but it makes it very difficult to write down.

In ZFC set theory, indexed family over a set (possibly uncountable or even bigger), is just syntactic sugar for a function.

So let's say you have a set U (possibly uncountable). To say let {u_i}, i in I (another set, possibly uncountable) is equivalent to asserting existence of function f:I -> U, such that f(i) = u_i. Note that this does not even require axiom of choice, since you are allowed to postulate that a function exists.

Of course if I is uncountable you can't list the elements of I, but that is not important in this case.

> It fixates on one particular basis and it results in a vector space with few applications and it can not explain many of the most important function vector spaces, which are of course the L^p spaces.

Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.

Haha, this works if you already know what a vector space is. But I think pedagogy needs to provide motivating examples. I'll quote one section of a text by Poincaré (translated by an LLM since most here do not speak French).

> We are in a geometry class. The teacher dictates: “A circle is the locus of points in the plane that are at the same distance from an interior point called the center.” The good student writes this sentence in his notebook; the bad student draws little stick figures in it; but neither one has understood. So the teacher takes the chalk and draws a circle on the board. “Ah!” think the students, “why didn’t he say right away: a circle is a round shape — we would have understood.”

> No doubt, it is the teacher who is right. The students’ definition would have been worthless, since it could not have served for any demonstration, and above all because it would not have given them the salutary habit of analyzing their conceptions. But they should be shown that they do not understand what they think they understand, and led to recognize the crudeness of their primitive notion, to desire on their own that it be refined and improved.

The learning comes from making the mistake and being corrected, not from being taught the definition, I think.

Anyway, it's from Science and Method, Book 2 https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre...

There's more to the section that talks about the subject. I just find this particular paragraph amusingly germane.

Completely agree. In uni, I (re)-learned about vectors in linear algebra, and for a good chunk of the course, we didn't write anything in "standard vector notation". We learned about vector axioms first, and then vectors were treated as "anything that satisfies the vector axioms". (When doing more practical examples, we just used the reals instead of something like R^3, but the entire time it was clear that for any proof, anything that can be added and multiplied in the way that the vector axioms describe would fit.) I think adopting this structuralist view really helps with a lot of mathematical studies.

Now I’m thinking that I have missed the point of the article. I didn’t read it as an introduction to vector spaces, but rather that the introduction served as to give an intuition how functions may be viewed as vectors (going back to the article, it’s even in the section heading). I found the next parts well written and to the point, leading along the steps to show that indeed the requirements for a Hilbert space are met by L^2 (even though those requirements are only spelled out in the end). I’m not actively working with mathematics any more, but I didn’t notice any major corner cutting. It’s not text book rigorous but lays out the idea in an easy to follow way. I took something away from it - or not, depending on whether I missed some inconsistency.

> In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is.

Could you spell out what you mean by that? Functions are all defined on their domains (by definition)

Are you referring to the L^p spaces being really equivalence classes of functions agreeing almost everywhere?

reminded me of "tensor is a bunch of numbers that transform in a certain way"; this should be illegal to teach, especially in physics

> In most function vector spaces you encounter in mathematics, you can not say what the value of a function at a point is. They are not defined that way.

That is because they are not vector spaces of function but a quotient of one

ianam but I'm not sure there needs to be a "why" -- these things are defined to have certain properties, and so they have those properties. That's the purpose for which they were defined.

You are observing the evaluation map ev_p: C(M, R) -> R, ev_p(f) = f(p), is a ring homomorphism. In that spirit, you might find this Terry Tao article on the Yondea lemma interesting: https://terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-a...

Let T^S = {f| f:S->T} be all functions from a source set S to a target structure (set + operations + relations + axioms) T.

You can lift all the operations and relations from T to T^S, and you'll get a structure with the same type signature.

Universal equations involving operations remain true when lifted. Therefore if T is a variety[0], T^S is a variety of the same type.

So for example if S a set with 2 elements, then T^S is TxT + lifted properties. If T is an Abelian group then TxT is also an Abelian group. If T is a ring, TxT is also a ring. If T is a field, TxT is not a field since (0,1) has no inverse.

What about the relations, what types of identities remain true when lifted form T to T^S?

[0] : https://en.wikipedia.org/wiki/Variety_(universal_algebra)