Definition 17.1.2 (Norming Duality).label Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality of normed vector spaces over $K$, then $\dpn{E, F}{\lambda}$ is norming if:

1. (1)

   For each $x \in E$, $\norm{x}_{E} = \sup_{y \in F, \norm{y}_F \le 1}\dpn{x, y}{\lambda}$.

2. (2)

   For each $y \in F$, $\norm{y}_{F} = \sup_{x \in E, \norm{x}_E \le 1}\dpn{x, y}{\lambda}$.