Functional analysis is just linear algebra + topology + a healthy respect for infinity. If you understand $\mathbbR^n$ and limits, you already have 80% of the intuition.
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
Why does $x = (1,1,1,\dots)$ cause trouble when multiplied by the matrix above? (Answer: The first component becomes the harmonic series, which diverges.) 1.3 From Solving Equations to Finding Functions The core idea of functional analysis is this: Functional analysis is just linear algebra + topology
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. This includes the Title Page, Table of Contents,
Bridging the gap from linear algebra to infinite-dimensional spaces without the fear factor