i have an exercise where i have to decide if given statements are right or wrong, and i am really frustrated by this one: Given a normed K-linear space X, the question is: Is it right that the dual X* is finite-dimensional if and only if the closed unit ball in X is compact with respect to the norm topology? I could not find any helpful theorems in the script of my lecture at all and intuitively i don`t really have a good idea why X* being finite-dimensional should depend on the closed unit ball in X being compact. Is there anyone who could help me to develop an idea of why these statements should depend on each other at all?
Thanks in advance!
