TITLE: Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
QUESTION [70 upvotes]: 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, infrabarrelled, Montel, reflexive, separable, Schwartz, convenient, semi-reflexive, reflexive, $c^\infty$-top is LCS top, quasi-complete, complete, Baire, nuclear

Its dual space (with the strong topology) is

reflexive, semi-reflexive, barrelled, infrabarrelled, 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?".

REPLY [18 votes]: Apology of one maker of a slight mess:
Whether properties are front line or supporting depends very much on which war you are fighting. If you are interested in the linear locally convex theory then many properties
are front line. But if you are mainly interested in smooth mappings, then quite few properties are front line; especially those which allow for uniform boundedness theorems (Greg should have added webbed in the diagram).
In fact, the space of smooth curves in a lcs (= locally convex space) does not change if you change the topology, as long as the bornology stays the same. In fact you can start from a dual pair of spaces  $(E, E')$ (separating points on each other), check, whether Mackey Cauchy sequences for the weak topology converge (then $E$ is convenient), and then you can choose any lcs topology which is compatible with the duality. The finest such topology is bornological. This was part of the approach in the book of Froelicher and Kriegl, who started from scratch and reconstructed everything. 
The most natural topology on $E$ from the point of view of Calculus is the final topology with respect to all smooth curves (equiv: all Mackey convergent sequences, equiv: all locally Lipschitz curves, ...); this is denoted $c^\infty E$. The finest lcs topology coarser than $c^\infty E$ is the bornologification of any lcs toplogy which is compatible with dual pair.  
Or you can try to allow users to make use of knowledge in the theory of lcs. This was the approach in the book: Convenient setting ... 
There you are allowed to use any lcs topology on $E$ that you know well or can describe well and which still has the given bornology. Spaces $E$ are identified if they are 
biboundedly linearly isomorphic (equiv.: diffeomorphic). To be convenient is a property of such equivalence classes of spaces. 
[Added in edit:] Or, to be convienient is a property shared by all spaces in such an equivalence class (or a space with all lcs topologies with the same system of bounded sets). The relation to the Frölicher-Kriegl notion is: take the bornologification of the space in question, as in their book only bornological spaces are considered. 
So, in Greg's nice diagram above, the place of convenient should be:
Sequentially complete $\implies$ convenient. 
In fact, each "naturally described" lcs is convenient. You have to force it to be not convenient by choosing a not Mackey complete subspace with the induced lcs topology.