I would like to find an elementary proof of the following theorem
Let $E$ be a normed space. Then the following statements are equivalent:
(a) E is finite dimensional.
(b) Every linear functional on $E$ is continuous.
(c) Every linear subspace of $E$ is closed.
I would like to prove $(a)\Leftrightarrow (b)$ and $(b)\Leftrightarrow (c)$
Thank you for all helping.