Continuous linear operator
- Continuous linear operator
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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 x0 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
- Rudin, Walter (January 1991). Functional Analysis. McGraw-Hill Science/Engineering/Math. ISBN 0-07-054236-8.
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