[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