That is what functional analysis does. It takes the geometric intuition of $\mathbbR^n$ and carefully extends it to infinite-dimensional spaces of functions.
Now, take a deep breath. Turn the page. Let's befriend functional analysis.
Here is the content for a book titled (PDF format). This includes the Title Page, Table of Contents, Preface, and a Sample Chapter (Chapter 1) to give you the structure and tone. TITLE PAGE A FRIENDLY APPROACH TO FUNCTIONAL ANALYSIS a friendly approach to functional analysis pdf
Bridging the gap from linear algebra to infinite-dimensional spaces without the fear factor
But here’s the secret the world didn't tell you: . That is what functional analysis does
Glossary of "Scary Terms" with Friendly Definitions
This book is different.
— Alex Rivera 1.1 A Tale of Two Spaces: Finite vs. Infinite Dimensions You already know linear algebra. In linear algebra, you work in $\mathbbR^n$ or $\mathbbC^n$. You have vectors $(x_1, x_2, \dots, x_n)$. You have matrices. You solve $Ax = b$. Life is good.