Continuous linear operator: Difference between revisions

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If <math>U \subseteq X</math> is a set then <math>F : X \to Y</math> is said to be {{em|bounded on <math>U</math>}} if <math>F(U)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y,</math> and it is said to be {{em|unbounded on <math>U</math>}} otherwise.

In particular, a map <math>F : X \to Y</math> is said to be {{em|{{visible anchor|bounded on a neighborhood of}}}} a point <math>x \in X</math> or {{em|{{visible anchor|locally bounded at}}}} <math>x</math> if there exists a [[Neighborhood (mathematics)|neighborhood]] <math>U</math> of this point in <math>X</math> such that <math>F(U)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y.</math> (in contrast, <math>F</math> is call {{em|[[Bounded linear operator|bounded]]}} if {{em|for every}} <math>B \subseteq X</math> that is a [[Bounded set (topological vector space)| {{em|bounded}}]] subset of <math>X,</math> its image <math>F(B)</math> is a bounded subset of <math>Y</math>).

A map is called [[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]] if it is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to be a synonym of "[[bounded linear operator]]", (which are related but {{em|not}} equivalent concepts (<math>F</math> is called {{em|[[Bounded linear operator|bounded]]}} if for every [[Bounded set (topological vector space)|bounded subset]] <math>B \subseteq X,</math> whichits image <math>F(B)</math> is whya bounded subset of <math>Y</math>). For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded {{em|at a point}}").

A linear map is locally bounded at every point of its ___domain if and only if there exists a point in its ___domain at which it is locally bounded; or said differently, if and only if it is "{{em|{{visible anchor|bounded on a neighborhood}}}}" (of some point).

A map is called [[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]] if it is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to be a synonym of "bounded linear operator" (which are related but {{em|not}} equivalent concepts), which is why this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded {{em|at a point}}").

If a linear map is locally bounded at {{em|some}} point then it is locally bounded at {{em|every}} point.

AThus a linear map is locally bounded at every point of its ___domain if and only if there exists a point in its ___domain at which it is locally bounded;, or said differently, if and only if it is "{{em|{{visible anchor|bounded on a neighborhood}}}}" (of some point).

A linear map that is bounded on a neighborhood (of some/every point) is necessarily continuous{{sfn|Narici|Beckenstein|2011|pp=156-175}} (even if its ___domain is not a [[normed space]]). The next example shows that the converse is not always guaranteed.