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.

If such an operator is between two normed spaces, then it is a bounded linear operator.

Properties

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A continuous linear operator maps bounded sets into bounded sets.

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

$A^\{-1\}(D)+x\_0=A^\{-1\}(D+Ax\_0)\; \backslash ,\backslash !$

for any set D in Y and any x₀ in X, which is true due to the additivity of A.

Functional analysis

Vikipedio:Projekto matematiko/Kontinua lineara operatoro