Editing Operator norm (section)

==Equivalent definitions==

Let <math>A : V \to W</math> be a linear operator between normed spaces. The first four definitions are always equivalent, and if in addition <math>V \neq \{0\}</math> then they are all equivalent:
:<math>
\begin{alignat}{4}
\|A\|_\text{op} &= \inf &&\{ c \geq 0 ~&&:~ \| A v \| \leq c \| v \| ~&&~ \text{ for all } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| \leq 1 ~&&~\mbox{ and } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| < 1 ~&&~\mbox{ and } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| \in \{0,1\}  ~&&~\mbox{ and } ~&&v \in V \} \\
&= \sup &&\{ \| Av \| ~&&:~ \| v \| = 1 ~&&~\mbox{ and } ~&&v \in V \}  \;\;\;\text{ this equality holds if and only if } V \neq \{ 0 \} \\
&= \sup &&\bigg\{ \frac{\| Av \|}{\| v \|} ~&&:~ v \ne 0 ~&&~\mbox{ and } ~&&v \in V \bigg\}  \;\;\;\text{ this equality holds if and only if } V \neq \{ 0 \}. \\
\end{alignat}
</math>
If <math>V = \{0\}</math> then the sets in the last two rows will be empty, and consequently their [[supremum]]s over the set <math>[-\infty, \infty]</math> will equal <math>-\infty</math> instead of the correct value of <math>0.</math> If the supremum is taken over the set <math>[0, \infty]</math> instead, then the supremum of the empty set is <math>0</math> and the formulas hold for any <math>V.</math> 

Importantly, a linear operator <math>A : V \to W</math> is not, in general, guaranteed to achieve its norm <math>\|A\|_\text{op} = \sup \{\|A v\| : \|v\| \leq 1, v \in V\}</math> on the closed unit ball <math>\{v \in V : \|v\| \leq 1\},</math> meaning that there might not exist any vector <math>u \in V</math> of norm <math>\|u\| \leq 1</math> such that <math>\|A\|_\text{op} = \|A u\|</math> (if such a vector does exist and if <math>A \neq 0,</math> then <math>u</math> would necessarily have unit norm <math>\|u\| = 1</math>). R.C. James proved [[James's theorem]] in 1964, which states that a [[Banach space]] <math>V</math> is [[reflexive space|reflexive]] if and only if every [[bounded linear functional]] <math>f \in V^*</math> achieves its [[Dual norm|norm]] on the closed unit ball.{{sfn|Diestel|1984|p=6}} 
It follows, in particular, that every non-reflexive Banach space has some bounded linear functional (a type of bounded linear operator) that does not achieve its norm on the closed unit ball.

If <math>A : V \to W</math> is bounded then{{sfn|Rudin|1991|pp=92-115}} 
<math display="block">\|A\|_\text{op} = \sup \left\{\left|w^*(A v)\right| : \|v\| \leq 1, \left\|w^*\right\| \leq 1 \text{ where } v \in V, w^* \in W^*\right\}</math>
and{{sfn|Rudin|1991|pp=92-115}}  
<math display="block">\|A\|_\text{op} = \left\|{}^tA\right\|_\text{op}</math>
where <math>{}^t A : W^* \to V^*</math> is the [[Transpose of a linear map|transpose]] of <math>A : V \to W,</math> which is the linear operator defined by <math>w^* \,\mapsto\, w^* \circ A.</math>