How to find a bounded linear functional which is nonzero at given point without using Hahn-Banach Theorem Let $X$ be a normed vector space. For any $0\neq x \in X$, there exists
$l \in X^*$ such that $l(x)\neq 0$. This statement can be easily proved based on the Hahn-Banach Theorem. My question is how to prove this without using the theorem. 

Here $X^{*}$ means the dual space of $X$. The dual space consists of all bounded linear functionals from $X$ to $\mathbb{R}$ or $\mathbb{C}$.
Take $0 \neq x \in X$ and complement $x$ by a family of vectors $(e_j)_{j \in J}$ to get a basis $(x,(e_j)_{j \in J})$ of $X$. This uses Axiom of choice.

Then define $l(x)=1$ and $l(e_j)=0$ for $j \in J$. $l$ satisfies the requested properties.
Take $0 \neq x \in X$ and complement $x$ by a family of vectors $(e_j)_{j \in J}$ to get a basis $(x,(e_j)_{j \in J})$ of $X$. This uses Axiom of choice.

Then define $l(x)=1$ and $l(e_j)=0$ for $j \in J$. $l$ satisfies the requested properties.