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==Continuity and boundedness==
'''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>
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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'''
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. ▼
{{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|[[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 <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>
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.
The next example shows that
▲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{{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}}
▲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]]".
{{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]]).
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Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
'''Guaranteeing that "continuous" implies "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.▼
'''Guaranteeing "bounded" implies "continuous"'''▼
Every [[sequentially continuous]] linear operator is bounded.{{sfn|Wilansky|2013|pp=47-50}} ▼
A linear map whose ___domain is [[Metrizable topological vector space|pseudometrizable]] is bounded if and only if it is continuous.{{sfn|Narici|Beckenstein|2011|pp=156-175}} ▼
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}}▼
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. ▼
But without additional information about either the linear map or it's ___domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood".▼
▲'''Guaranteeing "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.
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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 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}}
▲
'''Guaranteeing that "bounded" implies "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==
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