Editing Bounded operator (section)

{{Short description|Linear transformation between topological vector spaces}}
{{Distinguish|text=[[bounded function]] (set theory)}}
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In [[functional analysis]] and [[operator theory]], a '''bounded linear operator''' is a [[linear transformation]] <math>L : X \to Y</math> between [[topological vector space]]s (TVSs) <math>X</math> and <math>Y</math> that maps [[Bounded set (topological vector space)|bounded]] subsets of <math>X</math> to bounded subsets of <math>Y.</math> 
If <math>X</math> and <math>Y</math> are [[normed vector space]]s (a special type of TVS), then <math>L</math> is bounded if and only if there exists some <math>M > 0</math> such that for all <math>x \in X,</math>
<math display=block>\|Lx\|_Y \leq M \|x\|_X.</math>
The smallest such <math>M</math> is called the [[operator norm]] of <math>L</math> and denoted by <math>\|L\|.</math> 
A linear operator between normed spaces is [[Continuous linear operator|continuous]] if and only if it is bounded.

The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces.

Outside of functional analysis, when a function <math>f : X \to Y</math> is called "[[Bounded function|bounded]]" then this usually means that its [[Image of a function|image]] <math>f(X)</math> is a bounded subset of its codomain. A linear map has this property if and only if it is identically <math>0.</math> 
Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in this abstract sense (of having a bounded image).