Earlier quoted context omitted.
I strongly disagree. It's necessary to teach oversimplified things to small minds, even oversimplified to the extent of being incorrect. It's unlikely you understand relativistic mechanics unless you learn and practice Newtonian for a few years, then many people stop there and never go for relativism. It's unlikely you understand Peano axioms without first spending years just working with numbers, completely disregar…
To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. Mine gave me it so well, I still see no difference between R^n and a Hilbert space! (on the dangers of intuitions) This reminds me of a short lived stretch in French mathematical education, the "Bourbaki years". For those unaware, Nicolas Bourbaki was "a dozen…
That's the thing: not only he didn't share this intuition, he actively prohibited people from bringing it up in the class because it can lead to mistakes.
The program was to start with most generic cases (Banach spaces, metric spaces) and prove whatever is provable there, then continue adding assumptions one by one and proving stronger and stronger theorems. I think we've reached Hilbert spaces by the end of the year and that was a gotcha moment for many students (wait, these vectors were functions the whole time?), but it was too late. Everyone failed miserably at proving or understanding all the preceding theorems because without the intuition it turns into a game of symbols with no structure or hope. The only recourse were those bootleg analogies from finite spaces and Fourier analysis that a few students knew or came up with.
The "New Math" you mention above gives a good frame of reference. A beautiful curriculum that's almost impossible to understand unless you already learned math the normal way -- a formally incorrect and inconsistent but reliable way.
I also made similar mistake myself when I tried to teach the C language to 12 y.o kids without saying the dreaded "just write it like that, you'll understand later". That experiment failed completely but I only understood why two years later when I myself was subjected to the functional analysis course. Or I think the complete realization came to me even later, with the C language experiment and the functional analysis story becoming pieces of the same puzzle.