I am trying to show that, for f a rapidly decreasing function, the mapping from f to f' is continuous. Since the space of RDF's is an F-space, Rudin says that I can use the closed graph theorem for F-spaces, and hence I need to show that the graph of the mapping is closed.
So I wish to show that is closed. I think that I can do this if it follows that if , then ?
Can someone tell me if this is true, and if so, how would I prove it?