That is what functional analysis does. It takes the geometric intuition of $\mathbbR^n$ and carefully extends it to infinite-dimensional spaces of functions.
Glossary of "Scary Terms" with Friendly Definitions a friendly approach to functional analysis pdf
Now, take a deep breath. Turn the page. Let's befriend functional analysis. That is what functional analysis does
The challenge: In infinite dimensions, not every Cauchy sequence converges unless you choose your space carefully. That's why we need and Hilbert spaces — they are the "complete" spaces where limits behave. a friendly approach to functional analysis pdf