Probably too late at this point for you, but in case other people come along… I’d recommend learning functional analysis first in the context of a theoretical mechanics course/textbook, rather than a math course/textbook. The physicists tend to do a better job explaining the intuitions (and give far more exposure to applications), which I find is the most important thing for a first exposure. Full rigorous detail is something you can pick up later, if and when you need it.
Personally I did the exact opposite, and found that very refreshing. Whenever I ran into a snippet of applied functional analysis without knowing the formal background it just confused me.
Probably too late at this point for you, but in case other people come along… I’d recommend learning functional analysis first in the context of a theoretical mechanics course/textbook, rather than a math course/textbook. The physicists tend to do a better job explaining the intuitions (and give far more exposure to applications), which I find is the most important thing for a first exposure. Full rigorous detail is something you can pick up later, if and when you need it.
Personally I did the exact opposite, and found that very refreshing. Whenever I ran into a snippet of applied functional analysis without knowing the formal background it just confused me.