My "canonical example" is Banach-Steinhaus in functional analysis: that, in nice locally convex topological spaces (Banach will do), weakly bounded (or pointwise bounded) implies bounded.
The machinery is quite technical, usually involving the Baire category theorem, but the result is very simple and very surprising. One especial point I like about this is that when you compare normed vector spaces with Banach spaces, then the process of adding more stuff (i.e. completion) actually limits the things that can go wrong. My intuition is that if you want to limit the bad behaviour then you need to work in smaller spaces rather than larger.
