I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, I often get bamboozled by all the different types that there are. At one point, I made a little summary of the properties of my favourite space (smooth functions from the circle to Euclidean space) and found that it was:
metrisable, barrelled, bornological, Mackey, infrabarelledinfrabarrelled, Montel, reflexive, separable, Schwartz, convenient, semi-reflexive, reflexive, $c^\infty$-top is LCS top, quasi-complete, complete, Baire, nuclear
It's dual space (with the strong topology) is
reflexive, semi-reflexive, barelledbarrelled, infrabarelledinfrabarrelled, quasi-complete, complete, bornological, nuclear, Mackey, convenient, $c^\infty$ top = LCS top, Schwartz, Montel, separable, DF space
I get the impression that many of these properties (and there are more!) are not "front line" properties but rather are conditions that guarantee that certain Big Theorems (like uniform boundedness, or open mapping theorem) hold. But as an outsider of functional analysis, it's not always clear to me which are "front line" and which are "supporters".
So that's my question: which of these properties (and others that I haven't specified) are main properties and which have more of a supporting role?
I realise that there's a little vagueness there as to exactly where the division lies - but that's part of the point of the question! If pressed, I would refine it to "Which of these properties would you expect to find used outside functional analysis, and which are more part of the internal machine?".