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[edit]

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[edit]

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