Dual norm

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

Definition

Let $X$ be a normed vector space with norm $\|\cdot \|$ and let $X^{*}$ denote its continuous dual space. The dual norm of a continuous linear functional $f$ belonging to $X^{*}$ is the non-negative real number defined by any of the following equivalent formulas:

$${\begin{alignedat}{5}\|f\|&=\sup &&\{\,|f(x)|&&~:~\|x\|\leq 1~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|<1~&&~{\text{ and }}~&&x\in X\}\\&=\inf &&\{\,c\in [0,\infty )&&~:~|f(x)|\leq c\|x\|~&&~{\text{ for all }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|=1{\text{ or }}0~&&~{\text{ and }}~&&x\in X\}\\&=\sup &&\{\,|f(x)|&&~:~\|x\|=1~&&~{\text{ and }}~&&x\in X\}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\&=\sup &&{\bigg \{}\,{\frac {|f(x)|}{\|x\|}}~&&~:~x\neq 0&&~{\text{ and }}~&&x\in X{\bigg \}}\;\;\;{\text{ this equality holds if and only if }}X\neq \{0\}\\\end{alignedat}}$$

where $\sup$ and $\inf$ denote the supremum and infimum, respectively. The constant $0$ map is the origin of the vector space $X^{*}$ and it always has norm $\|0\|=0.$ If $X=\{0\}$ then the only linear functional on $X$ is the constant $0$ map and moreover, the sets in the last two rows will both be empty and consequently, their supremums will equal $\sup \varnothing =-\infty$ instead of the correct value of $0.$

Importantly, a linear function $f$ is not, in general, guaranteed to achieve its norm $\|f\|=\sup\{|fx|:\|x\|\leq 1,x\in X\}$ on the closed unit ball $\{x\in X:\|x\|\leq 1\},$ meaning that there might not exist any vector $u\in X$ of norm $$