Editing Operator (mathematics) (section)

== Bounded operators==
{{main|Bounded operator|Operator norm|Banach algebra}}

Let {{mvar|U}} and {{mvar|V}} be two vector spaces over the same [[ordered field]] (for example; <math>\mathbb{R} </math>), and they are equipped with [[norm (mathematics)|norm]]s. Then a linear operator from {{mvar|U}} to {{mvar|V}} is called '''bounded''' if there exists {{math|''c'' > 0}} such that <math display="block">\|\operatorname{A}\mathbf{x}\|_V \leq c\ \|\mathbf{x}\|_U </math> for every '''{{math|x}}''' in {{mvar|U}}.
Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of {{mvar|U}} and {{mvar|V}}: <math display="block">\|\operatorname{A}\| = \inf\{\ c : \|\operatorname{A}\mathbf{x}\|_V \leq c\ \|\mathbf{x}\|_U \}.</math>
In case of operators from {{mvar|U}} to itself it can be shown that
: <math display="inline">\|\operatorname{A}\operatorname{B}\| \leq \|\operatorname{A}\| \cdot \|\operatorname{B}\|</math>.{{efn| In this expression, the raised dot merely represents multiplication in whatever scalar field is used with  {{mvar|V}}&nbsp;. }} 

Any unital [[normed algebra]] with this property is called a [[Banach algebra]]. It is possible to generalize [[spectral theory]] to such algebras. [[C*-algebra]]s, which are [[Banach algebras]] with some additional structure, play an important role in [[quantum mechanics]].