[ttoinou]

But it's the article talking about vectors as a sequence of reals and having a basis, then extending that to infinite sequences of reals. The author is playing on multiple definitions of vector to produce a "woaw that's cool" effect, and that's bad maths

[Sharlin]

There is only one definition of "vector space" (up to isomorphism anyway), and that's what the author uses. You'll note that he doesn't talk about bases at all, the assumption of a basis is entirely in your mind. The entire point of the article is that the ℝ→ℝ function space is a vector space. A vector space is not required to have a basis, but assuming the axiom of choice, every vector space does have (at least) one, including that of ℝ→ℝ functions.

[ttoinou]

The choice of basis is very important for the applications. It doesnt matter that function could be vector spaces in theory without constructing a base, the article is about "hey look at functions they're as easy to operate in as this thing you already know with 3D vectors"

[MarkusQ]

You are misreading the article. At a guess, you have come into it "knowing what a vector is" from physics or computer science. Specifically, a vector is not "a sequence of reals", it is any object that obeys a certain set of properties. He is not "extending" your definition, he's trying to teach you the actual definition, which is more general than the examples you are used to.

(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.

[ttoinou]

I learned vector spaces in maths and I'm aware of your definition, that's how I learned it. I'm also aware that the same word can be used with different meanings depending on the context.

  (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).

Imagine now an article saying "complex numbers are just natural numbers, you can apply euclidean division to them the exact same way"