Assume X to be a real reflexive Banach space. Why are sequential topological notions topological notions ? (relatively to the weak topology on X and the weak star topology on X*) For ex : sequentially closed set is closed, sequentially compact set is compact, and so on.
For compact set : by using the Eberlein-Šmulian Theorem, we get the result.
Question : How can we proceed for closed sets and continuous mappings ?.
Thanks.