Existence of a Norming Functional

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

Let $X$ be a normed space and let $z_0\in X\setminus\{0\}$ .

Corollary: There exists a bounded linear functional $f:X\to\mathbb{K}$ such that

\|f\|=1\quad\text{and}\quad f(z_0)=\|z_0\|.

This is a direct consequence of separation of a point and a subspace by applying it to $Y=\{0\}$ and $x_0=z_0/\|z_0\|$ .