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Adding short description: "Function between topological vector spaces" |
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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.
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Throughout, <math>F : X \to Y</math> is a [[linear map]] between [[topological vector space]]s (TVSs).
'''Bounded
{{See also|Bounded set (topological vector space)}}
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]].
If the space happens to also be a [[normed space]] (or a [[seminormed space]])
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'''
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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}}
'''
{{See also|Local boundedness}}
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<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
</li>
<li><math>f</math> is a [[#locally bounded|locally bounded at every point]] of its ___domain.</li>
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* {{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|
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|1999|p=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->
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