Dual norm explained

In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.

Definition

Let

\| ⋅ \|

and let

X^*

denote its continuous dual space. The dual norm of a continuous linear functional

belonging to

X^*

is the non-negative real number defined by any of the following equivalent formulas:

\begin\| f \| &= \sup &&\ \\ &= \sup &&\ \\ &= \inf &&