In a finite-dimensional space any dense convex subset is the whole space, and (by the Stone-Weierstrass theorem) there are many examples of dense convex cones which are subspaces in infinite-dimensional spaces.
What I wish to know is whether there is a dense convex cone which is not a linear subspace in a (infinite-dimensional) topological vector space?
And if so, can we identify the spaces (sufficient or necessary conditions) for which there are no such cones? (meaning every dense convex cone is a linear subspace)
In particular, I am interested in the cases where the space is
- The space of measures (or more generally-)
- A dual space of a Banach space (or even more generally-)
- A locally convex space.