Continuous linear operator: Difference between revisions

Revision as of 00:59, 18 February 2022

Revision as of 01:03, 18 February 2022

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===Continuity and boundedness===

{{See also|Local boundedness|Bounded linear operator|Bounded set (topological vector space)}}

Throughout, <math>F : X \to Y</math> is a linear map between [[topological vector space]]s (TVSs).

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'''"Continuous" and "bounded" but not "bounded on a neighborhood"'''

{{See also|Local boundedness}}

A linear map is "[[#bounded on a neighborhood|bounded on a neighborhood]]" (of some point) if and only if it is locally bounded at every point of its ___domain, in which case it is necessarily continuous{{sfn|Narici|Beckenstein|2011|pp=156-175}} (even if its ___domain is not a [[normed space]]) and thus also a [[bounded linear operator|bounded]] (because a continuous linear operator is always a [[bounded linear operator]]).{{sfn|Narici|Beckenstein|2011|pp=441-457}}