> 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 on

> 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.