Continuous linear operator: Difference between revisions

Revision as of 19:14, 8 November 2022

Revision as of 19:15, 8 November 2022

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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.

AngAnd 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"'''