Norming Functionals in a Separable Real Normed Space, and the Dual Separates Points

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $X$ be a separable real normed space with zero vector $0_{X}$, dual space $X^{*}$ and dual norm $\lVert\cdot\rVert_{*}$.

(i) Norming functional. For every $x\in X$ there is $F\in X^{*}$ with

$$\lVert F\rVert_{*}\le1\qquad\text{and}\qquad F(x)=\lVert x\rVert .$$

(ii) Unit norm. If $X\neq\{0_{X}\}$, then for every $x\in X$ there is $F\in X^{*}$ with $\lVert F\rVert_{*}=1$ and $F(x)=\lVert x\rVert$.

(iii) Separation of points. For all $x,y\in X$ with $x\neq y$ there is $F\in X^{*}$ with $F(x)\neq F(y)$.