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.