Hilbert Space is reflexive

A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$.

Let $X$ be a Hilbert space, would you help me to show that $X$ is reflexive.

One of the example is $L^2[a,b]$, the reason is its dual is $L^2$ and the second dual is $L^2$ again.

Hint: use the Riesz representation theorem twice.

Here's my answer, would you help me to check it.

Since $X^*$ is Hilbert space too, then for $g$ a bounded linear functional on $X^*$ there is $f_g \in X^*$ such that $g(f)=\langle f,f_g\rangle$ and $\|g\|=\|f_g\|$ (Riesz Representation Theorem). Since $f_g\in X^*$ then by Riesz Representation Theorem again there exist $x_{f_g}$ such that $f_g(x)=\langle x,x_{f_g}\rangle$ and $\|f_g\|=\|x_{f_g}\|$. Hence, $\|g\|=\|x_{f_g}\|$. So, $X^{**}=\{g_x:x\in X\}$