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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. | |||
An operator between two [[normed space]]s is a [[bounded linear operator]] if and only if it is a continuous linear operator. | |||
== Properties == | |||
A continuous linear operator maps [[Bounded set (topological vector space)|bounded set]]s into bounded sets. A [[linear functional]] is continuous if and only if its [[Kernel_(linear_operator)|kernel]] is closed. Every linear function on a finite-dimensional space is continuous. | |||
The following are equivalent: given a linear operator ''A'' between topological spaces ''X'' and ''Y'': | |||
# ''A'' is continuous at 0 in ''X''. | |||
# ''A'' is continuous at some point <math>x_0</math> in ''X''. | |||
# ''A'' is continuous everywhere in ''X''. | |||
The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality | |||
: <math>A^{-1}(D)+x_0=A^{-1}(D+Ax_0) \,\!</math> | |||
for any set ''D'' in ''Y'' and any ''x''<sub>0</sub> in ''X'', which is true due to the additivity of ''A''. | |||
==References== | |||
*{{cite book |last=Rudin |first=Walter |title=Functional Analysis |date=January 1991 |publisher=McGraw-Hill Science/Engineering/Math |isbn=0-07-054236-8}} | |||
{{Functional Analysis}} | |||
[[Category:Functional analysis]] | |||
Revision as of 19:56, 28 November 2013
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator.
Properties
A continuous linear operator maps bounded sets into bounded sets. A linear functional is continuous if and only if its kernel is closed. Every linear function on a finite-dimensional space is continuous.
The following are equivalent: given a linear operator A between topological spaces X and Y:
- A is continuous at 0 in X.
- A is continuous at some point in X.
- A is continuous everywhere in X.
The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality
for any set D in Y and any x0 in X, which is true due to the additivity of A.
References
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