Continuous linear operator: Difference between revisions

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{{Short description|Function between topological vector spaces}}

In [[functional analysis]] and related areas of [[mathematics]], a '''continuous linear operator''' or '''continuous linear mapping''' is a [[Continuous function (topology)|continuous]] [[linear transformation]] between [[topological vector space]]s.

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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Throughout, <math>F : X \to Y</math> is a [[linear map]] between [[topological vector space]]s (TVSs).

'''Bounded on a setsubset'''

{{See also|Bounded set (topological vector space)}}

The notion of a "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 {{em|[[Norm (mathematics)|norm]] bounded; that is}}, ifmeaning and only ifthat <math>\sup_{s \in S} \|s\| < \infty.</math>

A subset of a normed (or seminormed) space is called {{em|bounded}} if it is norm-bounded (or equivalently, von Neumann bounded).

For example, the scalar field (<math>\Reals</math> or <math>\Complex</math>) with the [[absolute value]] <math>|\cdot|</math> is a normed space, so a subset <math>S</math> is bounded if and only if <math>\sup_{s \in S} |s|</math> is finite, which happens if and only if <math>S</math> is contained in some open (or closed) ball centered at the origin (zero).

Any translation, scalar multiple, and subset of a bounded set is again bounded.

'''Function bounded on a set'''

If <math>S \subseteq X</math> is a set then <math>F : X \to Y</math> is said to be {{em|{{visible anchor|function bounded on a set|bounded on a set|text=bounded on <math>S</math>}}}} if <math>F(S)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y,</math> which if <math>(Y, \|\cdot\|)</math> is a normed (or seminormed) space happens 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 := \{x + s : s \in S\}</math> for every <math>x \in X</math> (because <math>F(x + S) = F(x) + F(S)</math> and any translation of a bounded set is again bounded) if and only if it is bounded on <math>c S := \{c s : s \in S\}</math> for every non-zero scalar <math>c \neq 0</math> (because <math>F(c S) = c F(S)</math> and any scalar multiple of a bounded set is again bounded).

Consequently, if <math>(X, \|\cdot\|)</math> is a normed or seminormed space, then a linear map <math>F : X \to Y</math> is bounded on some (equivalently, on every) non-degenerate open or closed ball (not necessarily centered at the origin, and of any radius) if and only if it is bounded on the closed unit ball centered at the origin <math>\{x \in X : \|x\| \leq 1\}.</math>

'''Bounded linear maps'''

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By definition, a linear map <math>F : X \to Y</math> between [[Topological vector space|TVS]]s is said to be {{em|[[Bounded linear operator|bounded]]}} and is called a {{em|{{visible anchor|bounded linear operator|text=[[bounded linear operator]]}}}} if for every [[Bounded set (topological vector space)|(von Neumann) bounded subset]] <math>B \subseteq X</math> of its ___domain, <math>F(B)</math> is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its ___domain. When the ___domain <math>X</math> is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if <math>B_1</math> denotes this ball then <math>F : X \to Y</math> is a bounded linear operator if and only if <math>F\left(B_1\right)</math> is a bounded subset of <math>Y;</math> if <math>Y</math> is also a (semi)normed space then this happens if and only if the [[operator norm]] <math>\|F\| := \sup_{\|x\| \leq 1} \|F(x)\| < \infty</math> is finite. Every [[sequentially continuous]] linear operator is bounded.{{sfn|Wilansky|2013|pp=47-50}}

'''BoundedFunction bounded on a neighborhood and local boundedness'''

{{See also|Local boundedness}}

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To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being [[bounded linear operator|bounded]], and being bounded on a neighborhood are all [[Logical equivalence|equivalent]].

A linear map whose ___domain {{em|or}} codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood.

AngAnd a [[bounded linear operator]] valued in a [[Locally convex topological vector space|locally convex space]] will be continuous if its ___domain is [[Metrizable topological vector space|(pseudo)metrizable]]{{sfn|Narici|Beckenstein|2011|pp=156-175}} or [[bornological space|bornological]].{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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

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</li>

<li><math>f</math> is continuous at the origin.

* By definition, <math>f</math> said to be continuous at the origin if for every open (or closed) ball <math>B_r</math> of radius <math>r > 0</math> centered at <math>0</math> in the codomain <math>\mathbb{F},</math> there exists some [[Neighbourhood (mathematics)|neighborhood]] <math>U</math> of the origin in <math>X</math> such that <math>f(U) \subseteq B_r.</math>
* If <math>B_r</math> is a closed ball then the condition <math>f(U) \subseteq B_r</math> holds if and only if <math>\sup_{u \in U} |f(u)| \leq r.</math>

** However,It assumingis important that <math>B_r</math> be a closed ball in this [[supremum]] characterization. Assuming that <math>B_r</math> is instead an open ball, then <math>\sup_{u \in U} |f(u)| < r</math> is a sufficient but {{em|not necessary}} condition for <math>f(U) \subseteq B_r</math> to be true (consider for example when <math>f = \operatorname{Id}</math> is the identity map on <math>X = \mathbb{F}</math> and <math>U = B_r</math>), whereas the non-strict inequality <math>\sup_{u \in U} |f(u)| \leq r</math> is instead a necessary but {{em|not sufficient}} condition for <math>f(U) \subseteq B_r</math> to be true (consider for example <math>X = \R, f = \operatorname{Id},</math> and the closed neighborhood <math>U = [-r, r]</math>). This is one of several reasons why many definitions involving linear functionals, such as [[polar set]]s for example, involve closed (rather than open) neighborhoods and non-strict <math>\,\leq\,</math> (rather than strict<math>\,<\,</math>) inequalities.

</li>

<li><math>f</math> is [[#bounded on a neighborhood|bounded on a neighborhood]] (of some point). Said differently, <math>f</math> is a [[#locally bounded at|locally bounded at some point]] of its ___domain.

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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 thusso 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>

Thus, ifIf <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.respectively, [[Bounded linear operator|bounded]]), or else all three are [[Discontinuous linear functional|discontinuous]] (resp.respectively, unbounded).

==Examples==

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Every linear map whose ___domain is a finite-dimensional Hausdorff [[topological vector space]] (TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.

Every (constant) map <math>X \to Y</math> between TVS that is identically equal to zero is a linear map that is continuous, bounded, and bounded on the neighborhood <math>X</math> of the origin. In particular, every TVS has a non-empty [[continuous dual space]] (although it is possible for the constant zero map to be its only continuous linear functional).

Suppose <math>X</math> is any Hausdorff TVS. Then {{em|every}} [[linear functional]] on <math>X</math> is necessarily continuous if and only if every vector subspace of <math>X</math> is closed.{{sfn|Wilansky|2013|p=55}} Every linear functional on <math>X</math> is necessarily 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}}

Suppose <math>X</math> is any Hausdorff TVS. Then {{em|every}} [[linear functional]] on <math>X</math> is necessarily continuous if and only if every vector subspace of <math>X</math> is closed.{{sfn|Wilansky|2013|p=55}} Every linear functional on <math>X</math> is necessarily 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}}

==Properties==

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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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* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1969|p=}} -->

* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->

* {{Cite book|last1=Rudin|first1=Walter|author-link1=Walter Rudin|isbn=978-0-07-054236-5|title=Functional analysis|yeardate=January 1991|publisher=McGraw-Hill Science/Engineering/Math|url-access=registration|url=https://archive.org/details/functionalanalys00rudi}} <!-- Rudin, Walter (1991) Functional Analysis -->

* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|1999|p=}} -->

* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->

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{{Topological vector spaces}}

[[Category:Continuous mappings]]

[[Category:Functional analysis]]

[[Category:Linear operators]]

[[Category:Operator theory]]

[[Category:Theory of continuous functions]]