Today on the way, my Chinese friend and I have discussed the following question: Let denote a convex subset of a locally convex topological linear space . Show that the closure of also is convex.
I have suggested the following solution.
We define as following . Then is continuous and since is convex. We then obtain that due to the continuity of , that is for every . Therefore is a convex set.
So what I am going to tell you is how correct the solution is? If no, what’s the problem, otherwise, what’s the main point? I will show you a little bit latter. I think I should go for sleep, it’s late now 😦