Editing Operator norm (section)

== Examples ==

Every real <math>m</math>-by-<math>n</math> [[matrix (mathematics)|matrix]] corresponds to a linear map from <math>\R^n</math> to <math>\R^m.</math> Each pair of the plethora of (vector) [[norm (mathematics)|norms]] applicable to real vector spaces induces an operator norm for all <math>m</math>-by-<math>n</math> matrices of real numbers; these induced norms form a subset of [[matrix norm]]s.

If we specifically choose the [[Euclidean norm]] on both <math>\R^n</math> and <math>\R^m,</math> then the matrix norm given to a matrix <math>A</math> is the [[square root]] of the largest [[eigenvalue]] of the matrix <math>A^{*} A</math> (where <math>A^{*}</math> denotes the [[conjugate transpose]] of <math>A</math>).<ref>{{Cite web|url=https://mathworld.wolfram.com/OperatorNorm.html|title=Operator Norm|last=Weisstein|first=Eric W.|authorlink = Eric W. Weisstein|website=mathworld.wolfram.com|language=en|access-date=2020-03-14}}</ref> 
This is equivalent to assigning the largest [[singular value]] of <math>A.</math>

Passing to a typical infinite-dimensional example, consider the [[sequence space]] <math>\ell^2,</math> which is an [[Lp space|L<sup>''p''</sup> space]], defined by
<math display="block">\ell^2 = \left\{ (a_n)_{n \geq 1} : \; a_n \in \Complex, \; \sum_n |a_n|^2 < \infty \right\}.</math>

This can be viewed as an infinite-dimensional analogue of the [[Euclidean space]] <math>\Complex^n.</math> 
Now consider a bounded sequence <math>s_{\bull} = \left(s_n\right)_{n=1}^\infty.</math> The sequence <math>s_{\bull}</math> is an element of the space <math>\ell^\infty,</math> with a norm given by
<math display="block">\left\|s_{\bull}\right\|_\infty = \sup _n \left|s_n\right|.</math>

Define an operator <math>T_s</math> by pointwise multiplication:
<math display="block">\left(a_n\right)_{n=1}^{\infty} \;\stackrel{T_s}{\mapsto}\;\ \left(s_n \cdot a_n\right)_{n=1}^{\infty}.</math>

The operator <math>T_s</math> is bounded with operator norm
<math display="block">\left\|T_s\right\|_\text{op} = \left\|s_{\bull}\right\|_\infty.</math>

This discussion extends directly to the case where <math>\ell^2</math> is replaced by a general <math>L^p</math> space with <math>p > 1</math> and <math>\ell^\infty</math> replaced by <math>L^\infty.</math>