Continuous linear operator: Difference between revisions

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{{See also|Local boundedness}}

In contrast, a map <math>F : X \to Y</math> is said to be {{em|{{visible anchor|bounded on a neighborhood of}}}} a point}}}} <math>x \in X</math> or {{em|{{visible anchor|locally bounded at}}}} <math>x</math> if there exists a [[Neighborhood (mathematics)|neighborhood]] <math>U</math> of this point in <math>X</math> such that <math>F(U)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y.</math>

It is "{{em|{{visible anchor|bounded on a neighborhood}}}}" (of some point) if there exists {{em|some}} point <math>x</math> in its ___domain at which it is locally bounded, in which case this linear map <math>F</math> is necessarily locally bounded at {{em|every}} point of its ___domain.

The term "[[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]]" is sometimes used to refer to a map that is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "[[bounded linear operator]]", which are related but {{em|not}} equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded {{em|at a point}}").