[Sharlin]

He does not assume anything! Any assumption is in your head only. Of course he starts from the specific type of vector spaces that's the most familiar to readers. But then he shows that there's nothing that requires a vector space to have a finite, or even countably infinite, dimension. What matters are the axioms.

[ttoinou]

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.

So the goal is to impress the reader by letting him believe we will easily apply our easy linear algebra to real -> real functions. But we can't.

[bmacho]

> So the goal is to impress the reader by letting him believe we will easily apply our easy linear algebra to real -> real functions. But we can't.

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.

[ttoinou]

What intuition can you gain ? This applies to any field, not just functions. You're gaining intuition for a more general concept and not what makes real->real functions different from others maths object