Continuous linear operator: Difference between revisions

Revision as of 23:37, 17 February 2022

Revision as of 00:02, 18 February 2022

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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>

IfIt is "{{em|{{visible anchor|bounded on a linearneighborhood}}}}" map(of issome locallypoint) boundedif atthere exists {{em|some}} point then<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.

AThe map is calledterm [[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]] ifis sometimes used to refer to a map itthat is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to 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}}").

For any linear map, if it is [[#bounded on a neighborhood|bounded on a neighborhood]] then it is continuous, and if it is continuous then it is [[Bounded linear operator|bounded]]. The converse statements are not true in general but they are both true when the linear map's ___domain is a [[normed space]]. Additional details are now given below.

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{{See also|Local boundedness}}

A linear map is locally bounded at every point of its ___domain if and only if there exists a point in its ___domain at which it is locally [[#bounded, oron saida differently, if and only if it is "{{em|{{visible anchorneighborhood|bounded on a neighborhood}}}}"]] (of some point).

A linear map that is bounded on a neighborhood (of some/every point) is necessarily continuous{{sfn|Narici|Beckenstein|2011|pp=156-175}} (even if its ___domain is not a [[normed space]]). The next example shows that the converse is not always guaranteed.