That's not really the assumption of the article, that's the assumption the article makes of the reader's knowledge of vectors as they begin reading. The article is trying to make the point that functions are also vectors, strictly speaking, and give some intuition as to what that means.

Basically 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.