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}}

Line 17 ⟶ 19:

</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>

</ol>

===Continuity and boundedness===

{{SeeThroughout, also|Local<math>F boundedness|Bounded: X \to Y</math> is a [[linear operator|Boundedmap]] setbetween ([[topological vector space]]s (TVSs)}}.

'''Bounded subset'''

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

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> 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</math> for every <math>x \in X</math> (because <math>F(x + S) = F(x) + F(S)</math>).

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

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|[[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.

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]].

In contrast, 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>

If the space happens to also be a [[normed space]] (or a [[seminormed space]]) then a subset <math>S</math> is von Neumann bounded if and only if it is {{em|[[Norm (mathematics)|norm]] bounded}}, meaning that <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'''

{{See also|Bounded linear operator}}

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}}

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

{{See also|Local boundedness}}

In contrast, a map <math>F : X \to Y</math> is said to be {{em|{{visible anchor|bounded on a neighborhood of a point|bounded on a neighborhood of the point|text=bounded on a neighborhood of}}}} a point <math>x \in X</math> or {{em|{{visible anchor|locally bounded at a point|text=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>

It is "{{em|{{visible anchor|bounded on a neighborhood}}}}" (of some point) if there exists {{em|some}} point <math>x</math> in its ___domain at which it is locally bounded, in which case this linear map <math>F</math> is necessarily locally bounded at {{em|every}} point of its ___domain.

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}}").

===Bounded on a neighborhood implies continuous implies bounded===

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}} 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.

A linear map is "[[#bounded on a neighborhood|bounded on a neighborhood]]" (of some point) if and only if it is locally bounded at every point of its ___domain, in which case it is necessarily [[Continuous function (topology)|continuous]]{{sfn|Narici|Beckenstein|2011|pp=156-175}} (even if its ___domain is not a [[normed space]]) and thus also [[bounded linear operator|bounded]] (because a continuous linear operator is always a [[bounded linear operator]]).{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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

AFor any linear map, if it is "[[#bounded on a neighborhood|bounded on a neighborhood]]" (ofthen someit point)is ifcontinuous,{{sfn|Narici|Beckenstein|2011|pp=156-175}}{{sfn|Wilansky|2013|pp=54-55}} and only if it is locally bounded at every point of its ___domain, in whichcontinuous casethen it is necessarily[[Bounded continuouslinear operator|bounded]].{{sfn|Narici|Beckenstein|2011|pp=156441-175457}} (even ifThe itsconverse ___domainstatements isare not atrue [[normedin space]])general andbut thusthey alsoare aboth [[boundedtrue linearwhen operator|bounded]]the (because alinear continuousmap's linear operator___domain is always a [[boundednormed linear operatorspace]]).{{sfn|Narici|Beckenstein|2011|pp=441-457}} Examples and additional details are now given below.

The next example shows that a continuous linear map need not be bounded on a neighborhood and so also demonstrates that being "bounded on a neighborhood" is {{em|not}} always synonymous with being "[[Bounded linear operator|bounded]]".

====Continuous and bounded but not bounded on a neighborhood====

The next example shows that it is possible for a linear map to be [[Continuous function (topology)|continuous]] (and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is {{em|not}} always synonymous with being "[[Bounded linear operator|bounded]]".

{{em|'''Example''': A continuous and bounded linear map that is not bounded on any neighborhood}}: If <math>\operatorname{Id} : X \to X</math> is the identity map on some [[locally convex topological vector space]] then this linear map is always continuous (indeed, even a [[TVS-isomorphism]]) and [[Bounded linear operator|bounded]], but <math>\operatorname{Id}</math> is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in <math>X,</math> which [[Kolmogorov's normability criterion|is equivalent to]] <math>X</math> being a [[seminormable space]] (which if <math>X</math> is Hausdorff, is the same as being a [[normable space]]).

This shows that it is possible for a linear map to be continuous but {{em|not}} bounded on any neighborhood.

Indeed, this example shows that every [[Locally convex topological vector space|locally convex space]] that is not seminormable has a linear TVS-[[automorphism]] that is not bounded on any neighborhood of any point.

Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.

'''===Guaranteeing "bounded" implies "bounded on a neighborhood"'''converses===

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]].

If <math>F : X \to Y</math> is a bounded linear operator from a [[normed space]] <math>X</math> into some TVS then <math>F : X \to Y</math> is necessarily continuous; this is because any open ball <math>B</math> centered at the origin in <math>X</math> is both a bounded subset (which implies that <math>F(B)</math> is bounded since <math>F</math> is a bounded linear map) and a neighborhood of the origin in <math>X,</math> so that <math>F</math> is thus bounded on this neighborhood <math>B</math> of the origin, which (as mentioned above) guarantees continuity.

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

And 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 "boundedcontinuous" implies "continuousbounded on a neighborhood"'''

A TVS is said to be {{em|locally bounded}} if there exists a neighborhood that is also a [[Bounded set (topological vector space)|bounded set]].{{sfn|Wilansky|2013|pp=53-55}} For example, every [[Normed space|normed]] or [[seminormed space]] is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin.

Every [[sequentially continuous]] linear operator is bounded.{{sfn|Wilansky|2013|pp=47-50}}

If <math>B</math> is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood <math>B</math>).

So in particular, a continuous linear operator is always a bounded linear operator{{sfn|Narici|Beckenstein|2011|pp=441-457}} but in general, a bounded linear operator need not be continuous.

AConsequently, a linear map whosefrom ___domaina islocally [[Metrizablebounded topologicalTVS vectorinto space|pseudometrizable]]any other TVS is boundedcontinuous if and only if it is continuous.{{sfn[[#bounded on a neighborhood|Narici|Beckenstein|2011|pp=156-175}}bounded on a neighborhood]].

Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if <math>X</math> is a TVS such that every continuous linear map (into any TVS) whose ___domain is <math>X</math> is necessarily bounded on a neighborhood, then <math>X</math> must be a locally bounded TVS (because the [[identity function]] <math>X \to X</math> is always a continuous linear map).

Every [[bounded linear operator]] from a [[bornological space]] into a [[Locally convex topological vector space|locally convex space]] is continuous.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood.{{sfn|Wilansky|2013|pp=53-55}}

Importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be "[[Bounded linear operator|bounded]]" (meaning that it is a [[bounded linear operator]]) but to {{em|not}} be continuous.

Conversely, if <math>Y</math> is a TVS such that every continuous linear map (from any TVS) with codomain <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=53-55}}

When the ___domain is [[Metrizable topological vector space|metrizable]] or [[Bornological space|bornological]], such as when it is a [[normed space]], and the codomain is [[Locally convex topological vector space|locally convex]], then a linear operator being "[[Bounded linear operator|bounded]]" is equivalent to it being continuous.{{sfn|Narici|Beckenstein|2011|pp=441-457}}

ButIn withoutparticular, additional information about either thea linear mapfunctional oron it'sa ___domainarbitrary orTVS codomain,is thecontinuous mapif beingand "bounded" is not equivalentonly toif it beingis "bounded on a neighborhood".{{sfn|Wilansky|2013|pp=53-55}}

Thus when the ___domain {{em|or}} the codomain of a linear map is normable or seminormable, then continuity will be [[Logical equivalence|equivalent]] to being bounded on a neighborhood.

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

'''Guaranteeing that "bounded" implies "continuous"'''

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, a linear map from a [[Normed space|normed]] or [[Seminormed space|seminormed]] space into a TVS 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.

A continuous linear operator is always a [[bounded linear operator]].{{sfn|Narici|Beckenstein|2011|pp=441-457}}

=== Properties of continuous linear operators ===

But importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be [[Bounded linear operator|bounded]] but to {{em|not}} be continuous.

A linear map whose ___domain is [[Locally convexMetrizable topological vector space|locally convexpseudometrizable]] [[metrizable(such topologicalas vectorany [[normed space]]) is [[normableBounded linear operator|bounded]] if and only if every linear functional on it is continuous.{{sfn|Narici|Beckenstein|2011|pp=156-175}}

The same is true of a linear map from a [[bornological space]] into a [[Locally convex topological vector space|locally convex space]].{{sfn|Narici|Beckenstein|2011|pp=441-457}}

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

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

In general, without additional information about either the linear map or its ___domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood".

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

If <math>F : X \to Y</math> is a bounded linear operator from a [[normed space]] <math>X</math> into some TVS then <math>F : X \to Y</math> is necessarily continuous; this is because any open ball <math>B</math> centered at the origin in <math>X</math> is both a bounded subset (which implies that <math>F(B)</math> is bounded since <math>F</math> is a bounded linear map) and a neighborhood of the origin in <math>X,</math> so that <math>F</math> is thus bounded on this neighborhood <math>B</math> of the origin, which (as mentioned above) guarantees continuity.

<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==

Line 128 ⟶ 154:

</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.

Line 136 ⟶ 163:

</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>

Line 151 ⟶ 178:

</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>

Line 177 ⟶ 204:

<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==

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

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.

<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>

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).

<li>If <math>X</math> is a TVS, then every bounded linear functional on <math>X</math> is continuous 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>

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==

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===

Line 194 ⟶ 232:

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

Line 202 ⟶ 240:

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==

* {{annotated link|Bounded linear operator}}

* {{annotated link|Compact operator}}

* {{annotated link|Continuous linear extension}}

* {{annotated link|Contraction (operator theory)}}

* {{annotated link|Discontinuous linear map}}

* {{annotated link|Finest locally convex topology}}

Line 215 ⟶ 258:

* {{annotated link|Positive linear functional}}

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

* {{annotated link|Topological vector space}}

* {{annotated link|Unbounded operator}}

Line 232 ⟶ 274:

* {{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=}} -->

Line 239 ⟶ 281:

{{Banach spaces}}

{{Functional Analysisanalysis}}

{{Topological vector spaces}}

{{TopologicalVectorSpaces}}

[[Category:Continuous mappings]]

[[Category:Functional analysis]]

[[Category:Linear operators]]

[[Category:Operator theory]]

[[Category:Theory of continuous functions]]