Continuous linear operator: Difference between revisions

Revision as of 02:08, 18 February 2022

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The notion of "bounded set" for a topological vector space is that of being a [[Bounded set (topological vector space)|von Neumann bounded set]].

If the space happens to also be a [[normed space]] (or a [[seminormed space]]), such as the scalar field with the [[absolute value]] for instance, then a subset <math>S</math> is von Neumann bounded if and only if it is [[Norm (mathematics)|norm]] bounded; that is, if and only if <math>\sup_{s \in S} \|s\| < \infty.</math>

If <math>S \subseteq X</math> is a set then <math>F : X \to Y</math> is said to be {{em|bounded on <math>S</math>}} if <math>F(S)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y,</math> andwhich itif <math>(Y, \|\cdot\|)</math> is saida tonormed be(or {{em|unboundedseminormed) onspace happens if and only if <math>\sup_{s \in S} \|F(s)\| < \infty.</math>}} otherwise.

If <math>(Y, \|\cdot\|)</math> is a normed (or seminormed) space then <math>F</math> is bounded on <math>S</math> if and only if <math>\sup_{s \in S} \|F(s)\| < \infty.</math>

A linear map <math>F</math> is bounded on a set <math>S</math> if and only if it is bounded on <math>x + S</math> for every <math>x \in X</math> (because <math>F(x + S) = F(x) + F(S)</math>).