Definition: Let $X$ be a topological vector space and let $x\in X$. Then $x$ defines a linear functional $\hat{x}$ on $X^*$ via $\hat{x}(f)=f(x)$ $(f\in X^*)$.
From this definition we can easily get $\hat{x}$ is weak* continuous on $X^*$. Therefore, we have $\hat{x}\in (X^*,wk^*)^*$.
Now let $X$ be a normed space. My question is from $\hat{x}\in (X^*,wk^*)^*$ how can we get $\hat{x}\in X^{**}$ without using $(X^*,wk^*)^*\subseteq X^{**}$?
If you mean how to show that $\hat x$ is bounded, it is simply $$ |\hat x(f)|=|f(x)|\leq\|f\|\,\|x\| $$