Assignment:
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.
Provide complete and step by step solution for the question and show calculations and use formulas.