Two exercises on convex cones in Banach spaces

I happened to work in a particular kind of finite dimensional algebras for quite a bit, and it's amazing how quickly you forget the subtleties which can occur in infnite dimensions. Now that I resumed a previous work that I had done, I came across an hypothesis that I put in a theorem, and it took me a while to (re-)understand why this hypothesis was not always satisfied. Here is the thing in the form of two exercises (so that if I forget it again, I can just take a look back here !). The second gives a solution to the first, so if you want to do it the hard way, do the first one first !

 

1) Find a closed convex cone C in a (real) Banach space V such that C-C is a strict subspace of V which is dense in V.

2) Let V be the Banach space of continuous functions on [0;1] with the uniform norm. Let I be the closed convex cone of non-decreasing elements of V.

a) Show that I-I is not V.

b) Show that I-I is dense in V.

You can show 2b by an abstract structural argument... or you can show the density inside a particular subspace of V that you already know to be dense in V.