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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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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
'''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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* {{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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