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Continuous linear functionals

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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}}

Suppose that <math>F : X \to Y</math> is a [[linear operator]] between two [[topological vector space]]s (TVSs).
The following are equivalent:
<ol>
<li><math>F</math> is continuous.</li>
<li><math>F</math> is [[Continuity at a point|continuous at some point]] <math>x \in X.</math></li>
<li><math>F</math> is continuous at the origin in <math>X.</math></li>
</ol>

If <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>

If <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>

If <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>

If <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>

If <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>

If <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>

If <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==

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

'''Bounded on a set'''

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

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|{{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</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).

'''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}}

'''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===

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}}

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====

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 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]].
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 "continuous" implies "bounded 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.
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>).
Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is [[#bounded on a neighborhood|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).

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}}
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}}
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=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 that "bounded" implies "continuous"'''

A continuous linear operator is always a [[bounded linear operator]].{{sfn|Narici|Beckenstein|2011|pp=441-457}}
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 [[Metrizable topological vector space|pseudometrizable]] (such as any [[normed space]]) is [[Bounded linear operator|bounded]] if and only if 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"'''

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".
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.

==Continuous linear functionals==
{{See also|Sublinear function}}

Every linear functional on a [[topological vector space]] (TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them.
However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.

===Characterizing continuous linear functionals===

Let <math>X</math> be a [[topological vector space]] (TVS) over the field <math>\mathbb{F}</math> (<math>X</math> need not be [[Hausdorff space|Hausdorff]] or [[Locally convex topological vector space|locally convex]]) and let <math>f : X \to \mathbb{F}</math> be a [[linear functional]] on <math>X.</math>
The following are equivalent:{{sfn|Narici|Beckenstein|2011|pp=126-128}}

<ol>
<li><math>f</math> is continuous.</li>
<li><math>f</math> is uniformly continuous on <math>X.</math></li>
<li><math>f</math> is [[Continuity at a point|continuous at some point]] of <math>X.</math>

</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>
** It is 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.
* Explicitly, this means that there exists some neighborhood <math>U</math> of some point <math>x \in X</math> such that <math>f(U)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>\mathbb{F};</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}} that is, such that <math display=inline>\displaystyle\sup_{u \in U} |f(u)| < \infty.</math> This supremum over the neighborhood <math>U</math> is equal to <math>0</math> if and only if <math>f = 0.</math>
* Importantly, a linear functional being "bounded on a neighborhood" is in general {{em|not}} equivalent to being a "[[bounded linear functional]]" because (as described above) it is possible for a linear map to be [[Bounded linear operator|bounded]] but {{em|not}} continuous. However, continuity and [[bounded linear functional|boundedness]] are equivalent if the domain is a [[Normed space|normed]] or [[seminormed space]]; that is, for a linear functional on a normed space, being "bounded" is equivalent to being "bounded on a neighborhood".
</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 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 so 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>
<li>The kernel of <math>f</math> is closed in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>Either <math>f = 0</math> or else the kernel of <math>f</math> is {{em|not}} dense in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=156-175}}</li>
<li>There exists a continuous seminorm <math>p</math> on <math>X</math> such that <math>|f| \leq p.</math>
* In particular, <math>f</math> is continuous if and only if the seminorm <math>p := |f|</math> is a continuous.</li>
<li>The graph of <math>f</math> is closed.{{sfn|Wilansky|2013|p=63}}</li>
<li><math>\operatorname{Re} f</math> is continuous, where <math>\operatorname{Re} f</math> denotes the [[real part]] of <math>f.</math></li>
</ol>

If <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>

If 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>

If 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>

If 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>
<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=451-457}}</li>
<li><math>f</math> is sequentially continuous at the origin.</li>
</ol>

and if in addition <math>X</math> is a vector space over the [[real numbers]] (which in particular, implies that <math>f</math> is real-valued) then this list may be extended to include:
<ol start=20>
<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>For any real <math>r,</math> the half-space <math>\{x \in X : f(x) \leq r\}</math> is closed.{{sfn|Narici|Beckenstein|2011|pp=225-273}}</li>
</ol>

If <math>X</math> is 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]] (respectively, [[Bounded linear operator|bounded]]), or else all three are [[Discontinuous linear functional|discontinuous]] (respectively, unbounded).

==Examples==

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}}

==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===

If <math>X</math> is a complex [[normed space]] and <math>f</math> is a linear functional on <math>X,</math> then <math>\|f\| = \|\operatorname{Re} f\|</math>{{sfn|Narici|Beckenstein|2011|p=128}} (where in particular, one side is infinite if and only if the other side is infinite).

Every non-trivial continuous linear functional on a TVS <math>X</math> is an [[open map]].{{sfn|Narici|Beckenstein|2011|pp=126-128}}
If <math>f</math> is a linear functional on a real vector space <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
<math display=block>f(U) := \{f(u) : u \in U\} \quad \text{ and } \quad |f(U)| := \{|f(u)| : u \in U\},</math>
the supremum <math>\,\sup_{u \in U} |f(u)|\,</math> can be written more succinctly as <math>\,\sup |f(U)|\,</math> because
<math display=block>\sup |f(U)| ~=~ \sup \{|f(u)| : u \in U\} ~=~ \sup_{u \in U} |f(u)|.</math>
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>B_{\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=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}}
* {{annotated link|Linear functionals}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Positive linear functional}}
* {{annotated link|Topologies on spaces of linear maps}}
* {{annotated link|Unbounded operator}}

==References==

{{reflist}}

* {{Adasch Topological Vector Spaces}} 
* {{Berberian Lectures in Functional Analysis and Operator Theory}} 
* {{Bourbaki Topological Vector Spaces}} 
* {{Conway A Course in Functional Analysis}} 
* {{cite book|last=Dunford|first=Nelson|title=Linear operators|publisher=Interscience Publishers|publication-place=New York|year=1988|isbn=0-471-60848-3|oclc=18412261|language=ro}} 
* {{Edwards Functional Analysis Theory and Applications}} 
* {{Grothendieck Topological Vector Spaces}} 
* {{Jarchow Locally Convex Spaces}} 
* {{Köthe Topological Vector Spaces I}} 
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} 
* {{Cite book|last1=Rudin|first1=Walter|author-link1=Walter Rudin|isbn=978-0-07-054236-5|title=Functional analysis|year=January 1991|publisher=McGraw-Hill Science/Engineering/Math|url-access=registration|url=https://archive.org/details/functionalanalys00rudi}} 
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} 
* {{Swartz An Introduction to Functional Analysis}} 
* {{Trèves François Topological vector spaces, distributions and kernels}} 
* {{Wilansky Modern Methods in Topological Vector Spaces}} 

{{Banach spaces}}
{{Functional analysis}}
{{Topological vector spaces}}

[[Category:Functional analysis]]
[[Category:Linear operators]]
[[Category:Operator theory]]
[[Category:Theory of continuous functions]]

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