Finite dimensional dual space

Question

Let $X$ be a normed linear space with a finite dimensional dual $X^*$. How do I prove X is also finite dimensional?

Answer 1

If $X^*$ is finite dimensional then so is $X^{**}$. Also, there is an embedding $X\subseteq X^{**}$ by $x\to \phi_x$ where $\phi_x(f)=f(x)$. A subspace of a finite dimensional space is finite dimensional.

Answer 2

If $\{f_i\}\quad i=1,2...n$

is a basis of $V^*$, than the set

$\{v_j\}\quad j=1,2...n$ such that $f_i(v_j)=\delta_{ij}$

is a basis for $V$

for $x\in V$ we have:

$x=\sum_{k=1}^nf_k(x)v_k$