It is just unhelpful in many ways. 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.
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.
The right didactic way, in my experience, is introducing vector spaces first. Vectors are elements of vector spaces, not because they can be written in any particular basis, but because they fulfill the formal definition. And because they fullfil the formal definition they can be written in a basis.
> 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.
I hesitate to call anything pedagogically "wrong" as people think and learn in different ways, but I think the coyness some teachers display about the vector space concept hampers and delays a lot of students' understanding.
Edit: Actually, I think the "start with 'concrete' lists of numbers and move to 'abstract' vector spaces" approach is misguided as it is based on the idea that the vector space is an abstraction of the lists of numbers, which I think is wrong.
The vector space and the lists of numbers are two equivalent, related abstractions of some underlying thing, eg. movements in Euclidean space, investment portfolios, pixel colours, etc. The difference is that one of the abstractions is more useful for performing numerical calculations and one better expresses the mathematical structure and properties of the entities under consideration. They're not different levels of abstraction but different abstractions with different uses.
I'd be inclined to introduce the one best suited to understanding first, or at least alongside the one used for computations. Otherwise students are just memorising algorithms without understanding, which isn't what maths education should be about, IMO. (The properties of those algorithms can of course be proved without the vector space concept, but such proofs are opaque and magical, often using determinants which are introduced with no better justification than that they allow these things to be proved.)
Functions are actually a great motivating example for the definition of a vector space, precisely because they are first look nothing like what student think of as a vector.
Except just about all relevant applications that exist in computer science and physics where fixating on a representation is the standard.
If you want to talk about applications, then this representation is especially bad. Since the intuition it gives is just straight up false.
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?
If you care about these you need something more restrictive, for example to study differential equations you can work in Sobolev spaces, where the continuity requirement allows you to identify an equivalent class with a well-defined function.
How can an uncountably infinite set be used as an index? I was fine with natural numbers (countably infinite) being an index obv, but a real seems a stretch. I get the mathematical definition of a function, but again, this feels like we suddenly lose the plot…
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.
> 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.
The point is that we need some way to deal with objects that are inherently infinite-dimensional.
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.
> and they aren’t irrational (i.e. they have a finite precision)
No, if you want only rational "indices", then your vector space has a countable basis. Interesting vector spaces in analysis are uncountably infinite dimensional. (And for this reason the usual notion of a basis is not very useful in this context.)
I'm not sure if I'm misunderstanding what you mean by 'finite precision' but the ordinary meaning of those words would seem to limit it to rational numbers?
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.
One way to think about it is that, even though you're defining an index that permits infinite amounts of subdivision, from any given house there's always a "next house up" in the vector: you can move up one space.
In a real-indexed vector, that notion doesn't apply. It's "infinity plus one" all the way down: whatever real value you pick to start with, x, there's no delta small enough to add to it such that there's no number between x and x+d.
Just to clarify, uncountability isn't necessary for this. It's true for the rational numbers too, which are countable.
The famous counterexample to all of this sort of thinking is Hilbert’s hotel, which I’m sure you know but want to point it out for people who haven’t seen it before because it’s pretty mind-blowing when you first encounter it.
Say you have a hotel with an infinite number of rooms numbered 1,2,3,… and so on and they are all occupied. A guest arrives- how do you accommodate them? Well you ask the person in room one to move to room 2, the person in room 2 to move to room 3, and in general the person in room n to move to room n+1. So every existing guest has a room and room 1 is now free for your new guest.
Ok but what if an infinite number of prospective guests arrive all at once and every room in your hotel is full. How do you accommodate them? Still no problem. You ask the guest in room 1 to move to room 2, the one in room 2 to move to room 4, and in general the guest in room n to move to room 2n. Now all your existing guests still have a room but you have freed up an infinite number of (odd-numbered) rooms for your infinite number of new guests to move into.
These are all countable infinities, and Cantor showed that if the number of rooms in your infinitely-roomed hotel is ℵ_0, then the number of real numbers is 2^ℵ_0, which is obviously quite a lot more.