Editing Operator norm (section)

== Introduction and definition ==

Given two normed vector spaces <math>V</math> and <math>W</math> (over the same base [[Field (mathematics)|field]], either the [[real number]]s <math>\R</math> or the [[complex number]]s <math>\Complex</math>), a [[linear map]] <math>A : V \to W</math> is continuous [[if and only if]] there exists a real number <math>c</math> such that<ref>{{Citation|last1=Kreyszig|first1=Erwin|title=Introductory functional analysis with applications|publisher=John Wiley & Sons|year=1978|isbn=9971-51-381-1|page=97}}</ref>
<math display="block">\|Av\| \leq c \|v\| \quad \text{ for all } v\in V.</math>

The norm on the left is the one in <math>W</math> and the norm on the right is the one in <math>V</math>. 
Intuitively, the continuous operator <math>A</math> never increases the length of any vector by more than a factor of <math>c.</math> Thus the [[Image (mathematics)|image]] of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as [[bounded operator]]s. 
In order to "measure the size" of <math>A,</math> one can take the [[infimum]] of the numbers <math>c</math> such that the above inequality holds for all <math>v \in V.</math> 
This number represents the maximum scalar factor by which <math>A</math> "lengthens" vectors. 
In other words, the "size" of <math>A</math> is measured by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of <math>A</math> as
<math display="block">\|A\|_\text{op} = \inf\{ c \geq 0 : \|Av\| \leq c \|v\| \text{ for all } v \in V \}.</math>

The infimum is attained as the set of all such <math>c</math> is [[Closed set|closed]], [[Empty set|nonempty]], and [[Bounded set|bounded]] from below.<ref>See e.g. Lemma 6.2 of {{harvtxt|Aliprantis|Border|2007}}.</ref>

It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces <math>V</math> and <math>W</math>.