Continuous linear operator: Difference between revisions

Revision as of 18:55, 8 November 2022

Revision as of 19:14, 8 November 2022

Line 95:

===Guaranteeing converses===

To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being [[bounded linear operator|bounded]], and being bounded on a neighborhood are all [[Logical equivalence|equivalent]].

A linear map whose ___domain {{em|or}} codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood.

Ang a [[bounded linear operator]] valued in a [[Locally convex topological vector space|locally convex space]] will be continuous if its ___domain is [[Metrizable topological vector space|(pseudo)metrizable]]{{sfn|Narici|Beckenstein|2011|pp=156-175}} or [[bornological space|bornological]].{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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

Line 107 ⟶ 111:

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=53-55}}

So in summary,Thus when the ___domain {{em|or}} the codomain of a linear map is normable or seminormable, then continuity will be [[Logical equivalence|equivalent]] to being bounded on a neighborhood.

'''Guaranteeing that "bounded" implies "continuous"'''