Existence of a Norming Functional

For any nonzero z0, there is a bounded functional f with ||f||=1 and f(z0)=||z0||.

Existence of a Norming Functional

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 $x_0=z_0/\|z_0\|$ .