Conceptualizing functions as infinite-dimensional vectors lets us apply the tools of linear algebra to a vast landscape of new problemsBasically it's an introduction to "functional analysis", the field of math that looks at functions like vectors. The term is in there sneakily, but this is really what the article is all about.
I had classes on this subject and we were encouraged to build upon linear algebra intuition (much like the author does), with some caveats (such as a bounded closed set is not necessarily compact in infinite dimension). Things break down, as the author hints at when they mention they didn't prove the Laplacian to be self-adjoint but only symmetric (this has to do with the operator's domain, a concern that doesn't exist for linear operators on finite dimensional spaces)... but it's still a very convenient way to think and talk about things, conceptually.
And he's starting from the assumption vectors are finite (cf. 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.
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.