Proving that a Banach space is of finite dimension

Question

Let $X$ be a Banach space with $\dim X\le \infty$. such that $X^*$ (the dual of $X$ ) is finite dimensional.

Then, show that $X$ is of finite dimensional too and $\dim X = \dim X^*$.

Note If wherenever we know that $X$ is of finite dimensision, it is easy using linear algebra tools to show that $\dim X=\dim X^*$.

Therefore the assumption here implies that $\dim X^*=\dim X^{**}$. Which clearly not what I want. Can anyone help?

Answer 1

Hint: $X$ is embedded in $X^{**}$ which is finite dimensional.

Answer 2

Hint: Use the canonical injection: $$\iota: X \to X^{**}: x \mapsto \mathrm{ev}_x.$$