Continuous linear operator: Difference between revisions

Revision as of 03:28, 18 February 2022

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'''Guaranteeing "continuous" implies "bounded on a neighborhood"'''

A TVS is said to be {{em|locally bounded}} if there exists a neighborhood of the origin that is [[Bounded set (topological vector space)|bounded]]. For example, every [[Normed space|normed]] or [[seminormed space]] is a locally bounded TVS since the unit ball centered at the origin is a bounded subset.

Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is [[#bounded on a neighborhood|bounded on a neighborhood]].{{sfn|Wilansky|2013|pp=54-55}}

Conversely, if <math>Y</math> is a TVS such that every continuous linear map (from any TVS) into <math>Y</math> is necessarily [[#bounded on a neighborhood|bounded on a neighborhood]], then <math>Y</math> must be a locally bounded TVS.{{sfn|Wilansky|2013|pp=54-55}}

In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.{{sfn|Wilansky|2013|pp=54-55}}

If <math>B</math> is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set.

FurthermoreConsequently, a linear map from alocally [[Normedbounded space|normed]]TVS orinto [[Seminormedany space|seminormed]] space into aother TVS is continuous if and only if it is bounded on a neighborhood.{{sfn|Wilansky|2013|pp=54-55}}

Thus when the ___domain or codomain of a linear map is normable or seminormable, then continuity is equivalent to being bounded on a neighborhood.