Learning Aid for Basic Theorems of Topological Vector Spaces in Functional Analysis
Solution 1:
If you want to understand well topological vector spaces (TVS), and learn all the basic theorems of functional analysis in their most abstract version, which is usually in the context of TVS and not of normed spaces, I am afraid there is no speedy way. (No royal way to geometry, as the ancient greeks said.)
I would recommend (without any resevations) the book of Walter Rudin, Functional Analysis, which builds the foundations of TVS in a rather elegant way. The books of G.M.H Köthe and H.H. Schaefer I wouldn't recommend for a first reading.