Continuous linear map

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.

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:

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.

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