Continuous linear operator: Difference between revisions

Revision as of 04:23, 18 February 2022

Revision as of 04:32, 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 also a [[Bounded set (topological vector space)|bounded set]].{{sfn|Wilansky|2013|pp=53-55}} 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 subsetneighborhood of the origin.

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 (so this map is thus bounded on this neighborhood <math>B</math>).

Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is [[#bounded on a neighborhood|bounded on a neighborhood]].