Show that (X, ||*||) is a Banach space if and only if {x in X: ||x||=1} is complete.
Know that in the first direction, we must show that {x in X: ||x||=1} is closed subset of X.
For the reverse direction, I know I have to take a cauchy sequence and translate it to the unit circle and then show that if it is convergent there, it is convergent outside of the unit circle.