Continuous linear operator: Difference between revisions

Revision as of 22:18, 15 December 2022

Revision as of 19:44, 11 January 2023

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An operator between two [[normed space]]s is a [[bounded linear operator]] if and only if it is a continuous linear operator.

== Continuous linear operators ==

{{See also|Continuous function (topology)|Discontinuous linear map}}

=== Characterizations of continuity ===

{{See also|Bounded operator}}

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</ol>

ifIf <math>Y</math> is [[Locally convex topological vector space|locally convex]] then this list may be extended to include:

<ol start=4>

<li>for every continuous [[seminorm]] <math>q</math> on <math>Y,</math> there exists a continuous seminorm <math>p</math> on <math>X</math> such that <math>q \circ F \leq p.</math>{{sfn|Narici|Beckenstein|2011|pp=126-128}}</li>

</ol>

ifIf <math>X</math> and <math>Y</math> are both [[Hausdorff space|Hausdorff]] locally convex spaces then this list may be extended to include:

<ol start=5>

<li><math>F</math> is [[weakly continuous]] and its [[transpose]] <math>{}^t F : Y^{\prime} \to X^{\prime}</math> maps [[Equicontinuity|equicontinuous]] subsets of <math>Y^{\prime}</math> to equicontinuous subsets of <math>X^{\prime}.</math></li>

</ol>

ifIf <math>X</math> is a [[sequential space]] (such as a [[Metrizable topological vector space|pseudometrizable space]]) then this list may be extended to include:

<ol start=6>

<li><math>F</math> is [[Sequential continuity at a point|sequentially continuous]] at some (or equivalently, at every) point of its ___domain.</li>

</ol>

ifIf <math>X</math> is [[Metrizable topological vector space|pseudometrizable]] or metrizable (such as a normed or [[Banach space]]) then we may add to this list:

<ol start=7>

<li><math>F</math> is a [[bounded linear operator]] (that is, it maps bounded subsets of <math>X</math> to bounded subsets of <math>Y</math>).{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>

</ol>

ifIf <math>Y</math> is [[seminormable space]] (such as a [[normed space]]) then this list may be extended to include:

<ol start=8>

<li><math>F</math> maps some neighborhood of 0 to a bounded subset of <math>Y.</math>{{sfn|Wilansky|2013|p=54}}</li>

</ol>

ifIf <math>X</math> and <math>Y</math> are both [[Normed space|normed]] or [[seminormed space]]s (with both seminorms denoted by <math>\|\cdot\|</math>) then this list may be extended to include:

<ol start=9>

<li>for every <math>r > 0</math> there exists some <math>\delta > 0</math> such that <math display=block>\text{ for all } x, y \in X, \text{ if } \|x - y\| < \delta \text{ then } \|F x - F y\| < r.</math></li>

</ol>

ifIf <math>X</math> and <math>Y</math> are Hausdorff locally convex spaces with <math>Y</math> finite-dimensional then this list may be extended to include:

<ol start=10>

<li>the graph of <math>F</math> is closed in <math>X \times Y.</math>{{sfn|Narici|Beckenstein|2011|p=476}}</li>

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</li>

<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 the origin and <math>\displaystyle\sup_{n \in N_r} |f(n)| = r.</math> Using <math>r := 1</math> proves the next statement when <math>R \neq 0.</math>

</li>

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

* This inequality holds if and only if <math>\sup_{x \in r U} |f(x)| \leq r</math> for every real <math>r > 0,</math> which shows that the positive scalar multiples <math>\{r U : r > 0\}</math> of this single neighborhood <math>U</math> will satisfy the definition of [[Continuity at a point|continuity at the origin]] given in (4) above.

* By definition of the set <math>U^{\circ},</math> which is called the [[Polar set|(absolute) polar]] of <math>U,</math> the inequality <math>\sup_{u \in U} |f(u)| \leq 1</math> holds if and only if <math>f \in U^{\circ}.</math> Polar sets, and thus also this particular inequality, play important roles in [[duality theory]].

</li>

<li><math>f</math> is a [[#locally bounded|locally bounded at every point]] of its ___domain.</li>

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</ol>

ifIf <math>X</math> and <math>Y</math> are complex vector spaces then this list may be extended to include:

<ol start=14>

<li>The imaginary part <math>\operatorname{Im} f</math> of <math>f</math> is continuous.</li>

</ol>

ifIf the ___domain <math>X</math> is a [[sequential space]] then this list may be extended to include:

<ol start=15>

<li><math>f</math> is [[Sequential continuity at a point|sequentially continuous]] at some (or equivalently, at every) point of its ___domain.{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>

</ol>

ifIf the ___domain <math>X</math> is [[Metrizable topological vector space|metrizable or pseudometrizable]] (for example, a [[Fréchet space]] or a [[normed space]]) then this list may be extended to include:

<ol start=16>

<li><math>f</math> is a [[bounded linear operator]] (that is, it maps bounded subsets of its ___domain to bounded subsets of its codomain).{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>

</ol>

ifIf the ___domain <math>X</math> is a [[bornological space]] (for example, a [[Metrizable topological vector space|pseudometrizable TVS]]) and <math>Y</math> is [[Locally convex topological vector space|locally convex]] then this list may be extended to include:

<ol start=17>

<li><math>f</math> is a [[bounded linear operator]].{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>

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<li>There exists a continuous seminorm <math>p</math> on <math>X</math> such that <math>f \leq p.</math>{{sfn|Narici|Beckenstein|2011|pp=126-128}}</li>

<li>For some real <math>r,</math> the half-space <math>\{x \in X : f(x) \leq r\}</math> is closed.</li>

<li>TheFor aboveany statementreal but with<math>r,</math> the wordhalf-space "some"<math>\{x replaced\in byX "any: f(x) \leq r\}</math> is closed."{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>

</ol>

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Every non-trivial continuous linear functional on a TVS <math>X</math> is an [[open map]].{{sfn|Narici|Beckenstein|2011|pp=126-128}}

Note that ifIf <math>Xf</math> is a reallinear vectorfunctional space, <math>f</math> ison a linearreal functionalvector onspace <math>X,</math> and if <math>p</math> is a seminorm on <math>X,</math> then <math>|f| \leq p</math> if and only if <math>f \leq p.</math>{{sfn|Narici|Beckenstein|2011|pp=126-128}}

If <math>f : X \to \mathbb{F}</math> is a linear functional and <math>U \subseteq X</math> is a non-empty subset, then by defining the sets

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If <math>s</math> is a scalar then

<math display=block>\sup |f(sU)| ~=~ |s| \sup |f(U)|</math>

so that if <math>r > 0</math> is a real number and <math>\overlineB_{B_r\leq r} := \{c \in \mathbb{F} : |c| \leq r\}</math> is the closed ball of radius <math>r</math> centered at the origin then the following are equivalent:

#<math display=inline>f(U) \subseteq B_{\leq 1}</math>

<math display=block>f(U) \subseteq \overline{B_1} \quad \text{ if and only if } \quad \sup |f(U)| \leq 1 \quad \text{ if and only if } \quad \sup |f(rU)| \leq r \quad \text{ if and only if } \quad f(rU) \subseteq \overline{B_r}.</math>

#<math display=inline>\sup |f(U)| \leq 1</math>

#<math display=inline>\sup |f(rU)| \leq r</math>

#<math display=inline>f(r U) \subseteq B_{\leq r}.</math>

==See also==

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* {{annotated link|Positive linear functional}}

* {{annotated link|Topologies on spaces of linear maps}}

* {{annotated link|Topological vector space}}

* {{annotated link|Unbounded operator}}

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{{Topological vector spaces}}

[[Category:Theory of continuous functions]]

[[Category:Functional analysis]]

[[Category:Linear operators]]

[[Category:Operator theory]]

[[Category:Theory of continuous functions]]