How Hahn-Banach theorem implies that the dual space is non-trivial? Why does the theorem of Hahn-Banach implies that the dual space is not empty ($X^*\neq\emptyset)$ ?
Is there an important corollary which I've missed ? 

The dual space is always non-empty, as it contains the zero functional. The Hahn-Banach theorem implies that if $X \neq \{0\}$, then also $X^* \neq \{0\}$. 
Choose a non-zero vector $a \in X$. Denote the subspace $Y := \mathrm{span} (a) \subseteq X$ and the bounded functional $\varphi \in Y^*$ defined by $\varphi(a) = 1$. By Hahn-Banach, it can be extended to some bounded functional on the whole space $\overline{\varphi} \in X^*$, which is non-zero (since $\overline{\varphi}(a) \neq 0$), hence $X^* \neq \{0\}$. 
