For the first question, put . Show that for all we have . Then use the fact that the maps and are continuous for the usual norm on .
I need help with this:
Let E a Banach Space and B(E) the Banach space of linear and bounded operators from E to E with the usual norm. Let S B(E).
If the sum convergs to the operator , show that = . And if T,S B(E) conmute then = . In particular prove that is invertible.
Please somebody give me a hand.