Continuous linear operator: Difference between revisions

Revision as of 04:45, 18 February 2022

Revision as of 22:33, 18 February 2022

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

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}}

The next example shows that a continuous linear map need not be bounded on a neighborhood and so also demonstrates that being "bounded on a neighborhood" is {{em|not}} always synonymous with being "[[Bounded linear operator|bounded]]".