Continuous linear operator: Difference between revisions

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===Continuity and boundedness===

{{See also|Local boundedness|Bounded linear operator|Bounded set (topological vector space)}}

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The term "[[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]]" is sometimes used to refer to a map that is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "[[bounded linear operator]]", which are related but {{em|not}} equivalent concepts. 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}}").

For any linear map, if it is [[#bounded on a neighborhood|bounded on a neighborhood]] then it is continuous,{{sfn|Narici|Beckenstein|2011|pp=156-175}}{{sfn|Wilansky|2013|pp=54-55}} and if it is continuous then it is [[Bounded linear operator|bounded]].{{sfn|Narici|Beckenstein|2011|pp=441-457}} The converse statements are not true in general but they are both true when the linear map's ___domain is a [[normed space]]. Examples and additional details are now given below.

'''"Continuous" and "bounded" but not "bounded on a neighborhood"'''

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

A TVS is said to be {{em|locally bounded}} if there exists a neighborhood of the origin that is [[Bounded set (topological vector space)|bounded]]. For example, every [[Normed space|normed]] or [[seminormed space]] is a locally bounded TVS.

A linear map from a TVS into a [[Normed space|normed]] or [[Seminormed space|seminormed]] space (such as a linear functional for example) is continuous if and only if it is bounded on some neighborhood. In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on some neighborhood.

In addition, aAny linear map from a [[NormedTVS space|normed]]into ora [[Seminormedlocally space|seminormed]]bounded spaceTVS into(such aas TVSany linear functional) is continuous if and only if it is [[#bounded on a neighborhood. Thus when the ___domain or codomain of a linear map is normable or seminormable, then continuity is equivalent to being |bounded on a neighborhood]].{{sfn|Wilansky|2013|pp=54-55}}

Conversely, if <math>Y</math> is a TVS such that every continuous linear map (from any TVS) into <math>Y</math> is necessarily [[#bounded on a neighborhood|bounded on a neighborhood]], then <math>Y</math> must be a locally bounded TVS.{{sfn|Wilansky|2013|pp=54-55}}

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=54-55}}

AFurthermore, a linear map from a TVS into a [[Normed space|normed]] or [[Seminormed space|seminormed]] space (such asinto a linear functional for example)TVS is continuous if and only if it is bounded on some neighborhood. In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on some neighborhood.{{sfn|Wilansky|2013|pp=54-55}}

=== Properties of continuous linear operators ===

Thus when the ___domain or codomain of a linear map is normable or seminormable, then continuity is equivalent to being bounded on a neighborhood.

A [[Locally convex topological vector space|locally convex]] [[metrizable topological vector space]] is [[normable]] if and only if every linear functional on it is continuous.

A continuous linear operator maps [[Bounded set (topological vector space)|bounded set]]s into bounded sets.

The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality

<math display=block>F^{-1}(D) + x = F^{-1}(D + F(x))</math>

for any subset <math>D</math> of <math>Y</math> and any <math>x \in X,</math> which is true due to the [[Additive map|additivity]] of <math>F.</math>

==Continuous linear functionals==

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Thus, if <math>X</math> is a complex then either all three of <math>f,</math> <math>\operatorname{Re} f,</math> and <math>\operatorname{Im} f</math> are [[Continuous linear map|continuous]] (resp. [[Bounded linear operator|bounded]]), or else all three are [[Discontinuous linear functional|discontinuous]] (resp. unbounded).

==Examples==

===Sufficient conditions for continuous linear functionals===

<li>Every linear functionmap onwhose ___domain is a finite-dimensional Hausdorff [[topological vector space]] (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.</li>

<ul>

<li>Every linear function on a finite-dimensional Hausdorff topological vector space (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.</li>

Suppose <limath>IfX</math> is any Hausdorff TVS. Then {{em|every}} [[linear functional]] on <math>X</math> is anecessarily TVS,continuous thenif and only if every boundedvector subspace of <math>X</math> is closed.{{sfn|Wilansky|2013|p=55}} Every linear functional on <math>X</math> is continuousnecessarily a bounded linear functional if and only if every [[Bounded set (topological vector space)|bounded subset]] of <math>X</math> is contained in a finite-dimensional vector subspace.{{sfn|Wilansky|2013|p=50}}</li>

</ul>

==Properties==

A [[Locally convex topological vector space|locally convex]] [[metrizable topological vector space]] is [[normable]] if and only if every bounded linear functional on it is continuous.

A continuous linear operator maps [[Bounded set (topological vector space)|bounded set]]s into bounded sets.

The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality

<math display=block>F^{-1}(D) + x = F^{-1}(D + F(x))</math>

for any subset <math>D</math> of <math>Y</math> and any <math>x \in X,</math> which is true due to the [[Additive map|additivity]] of <math>F.</math>

===Properties of continuous linear functionals===