Norm in terms of Normed Dual Space

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

Let $X^\ast$ be the normed dual of $X$.

Let $B_{X^\ast}^-$ be the closed unit ball of $X^\ast$.

Then:

$\ds \norm x = \sup_{f \in B_{X^\ast}^-} \cmod {\map f x}$

Proof

From Fundamental Property of Norm on Bounded Linear Functional, we have:

$\ds \cmod {\map f x} \le \norm x$

for each $x \in X$.

From Existence of Support Functional, there exists $f \in B_{X^\ast}^-$ such that $\map f x = \norm x$.

Hence we conclude:

$\ds \norm x = \sup_{f \in B_{X^\ast}^-} \cmod {\map f x}$

$\blacksquare$