Other nitpicks (which I don't think are real problems):

If the Wikipedia article on Kakatuni's fixed point theorem is to be believed, then Kakatuni's result is only for finite dimensional vector spaces. You probably want to be citing either Glicksberg or Fan for the infinite dimensional version. These each have some additional hypotheses, so you should check the additional hypotheses.

At the end of the proof of Theorem 2, you want to check that the graph of is closed. Let be the graph of . What you check is that, if is a sequence of points in which approaches a limit, then that limit is in . This set off alarm bells in my head, because there are examples of a topological space , and a subspace , so that is not closed in but, if is any sequence in which approaches a limit in , then that limit is in . See Wikipedia's article on sequential spaces. However, this is not an actual problem. Since is countable, is metrizable and therefore closure is the same as sequential closure in .