Prove that if $\dim X'<\infty$ then $\dim X<\infty$ I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$.
How can I prove this? I always try to figure out sth by myself before posting here, but this time I have no idea how can I prove this. 

Here's another approach. Again, we let $X'$ be the continuous dual.
Well, if $\dim X' < \infty$ then $\dim X'' = \dim X' < \infty$. But then note
$X \subset X''$ (using the canonical embedding) so $X$ is then finite dimensional!
