Is there a direct proof that a compact unit ball implies automatic continuity?Is there a direct proof that a compact unit ball implies automatic continuity?
We know that if a Banach space is infinitely dimensional, then there are discontinuous linear functionals on it. Therefore having only continuous functionals imply finite dimension (at least in the presence of choice).
At the same time, we know that a compact unit ball must imply finite dimension, and therefore every linear functional is continuous.
Is there a direct proof from the compactness of the unit ball, that every linear functional is continuous?
