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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}}").
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]].
'''"Continuous" and "bounded" but not "bounded on a neighborhood"'''
Every [[sequentially continuous]] linear operator is bounded.{{sfn|Wilansky|2013|pp=47-50}} ▼
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.▼
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}} ▼
{{See also|Local boundedness}}
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 a [[bounded linear operator|bounded]] (because a continuous linear operator is always a [[bounded linear operator]]).{{sfn|Narici|Beckenstein|2011|pp=441-457}}
{{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 "bounded" implies "continuous"'''
However, 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. ▼
▲Every [[sequentially continuous]] linear operator is bounded.{{sfn|Wilansky|2013|pp=47-50}}
▲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.
▲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}}
▲A linear map being "bounded on a neighborhood" is {{em|not}} the same as it being "[[Bounded linear operator|bounded]]".
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. ▼
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.
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}}
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"'''
▲
▲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.
'''Guaranteeing "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.
=== Properties of continuous linear operators ===
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