Continuous linear operator

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.

Definition

If for every $\epsilon >0$ there exists a $\delta >0$ such that $||x-y||<\delta \Rightarrow ||Ax-Ay||<\epsilon$ we say the operator $A$ is continuous.

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:

1. A is continuous at 0 in X.
2. A is continuous at some point $x_{0}$ in X.
3. 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

$A^{-1}(D)+x_{0}=A^{-1}(D+Ax_{0})\,\!$

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.