Continuous linear operator: Difference between revisions

Revision as of 22:21, 17 February 2022

Revision as of 22:22, 17 February 2022

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<li><math>f</math> is [[#bounded on a neighborhood of|bounded on a neighborhood of the origin]]. Said differently, <math>f</math> is a [[#locally bounded at|locally bounded at the origin.]]

* The equality <math>\sup_{x \in s U} |f(x)| = |s| \sup_{u \in U} |f(u)|</math> holds for all scalars <math>s</math> and when <math>s \neq 0</math> then <math>s U</math> will be neighborhood of the origin. So in particular, if <math display=inline>R := \displaystyle\sup_{u \in U} |f(u)|</math> is a positive real number then for every positive real <math>r > 0,</math> the set <math>N_r := \tfrac{r}{R} U</math> is also a neighborhood (of somethe point)origin and <math>\displaystyle\sup_{n \in N_r} |f(n)| = r.</math>

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<li>There exists some neighborhood <math>U</math> of the origin such that <math>\sup_{u \in U} |f(u)| \leq 1</math>