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===Continuity and boundedness===
Throughout, <math>F : X \to Y</math> is a linear map between [[topological vector
The notion of "bounded set" for a topological vector space
If a TVS happens to also be a normed (or seminormed) space then a subset <math>S</math> is von Neumann bounded if and only if it is norm bounded; that is, if and only if <math>\sup_{s \in S} \|s\| < \infty.</math>
If <math>
If <math>F</math> is a valued in a normed (or seminormed) space <math>(Y, |\cdot|),</math> such as <math>\R</math> or <math>\Complex</math> for instance, then it is bounded on a subset <math>S</math> if and only if <math>\sup_{s \in S} |F(s)| < \infty.</math>
By definition, a linear map <math>F : X \to Y</math> between [[
In
A map is called [[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]] if it is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to be a synonym of "[[bounded linear operator]]", which are related but {{em|not}} equivalent concepts
'''"Bounded" versus "continuous"
▲By definition, a linear map between [[topological vector space]]s (TVSs) is said to be {{em|[[Bounded linear operator|bounded]]}} and is called a {{em|[[bounded linear operator]]}} if it maps [[Bounded set (topological vector space)|(von Neumann) bounded subsets]] of the ___domain to bounded subsets of the codomain.
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.
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{{See also|Local boundedness}}
▲If <math>U \subseteq X</math> is a set then <math>F : X \to Y</math> is said to be {{em|bounded on <math>U</math>}} if <math>F(U)</math> is a [[Bounded set (topological vector space)|bounded subset]] of <math>Y,</math> and it is said to be {{em|unbounded on <math>U</math>}} otherwise.
▲In particular, 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>
▲A map is called [[Locally bounded function|{{em|{{visible anchor|locally bounded}}}}]] if it is locally bounded at every point of its ___domain, but some functional analysis authors define "locally bounded" to be a synonym of "[[bounded linear operator]]", which are related but {{em|not}} equivalent concepts (<math>F</math> is called {{em|[[Bounded linear operator|bounded]]}} if for every [[Bounded set (topological vector space)|bounded subset]] <math>B \subseteq X,</math> its image <math>F(B)</math> is a bounded subset of <math>Y</math>). 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}}").
If a linear map is locally bounded at {{em|some}} point then it is locally bounded at {{em|every}} point.
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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".
▲In summary, for any linear map, if it is bounded on a neighborhood then it is continuous, and if it is continuous then it is [[Bounded linear operator|bounded]]. The converse statements are not true in general but they are both true when the linear map's ___domain is a [[normed space]].
=== Properties of continuous linear operators ===
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