nLab
bounded operator

Contents

Idea

In a (Hausdorff) topological vector space one can consider bounded sets: a set SS is bounded if it is absorbed by any open neighborhood UU of zero (i.e. a dilated multiple λU={λx|xU}\lambda U = \{\lambda x | x\in U\} contains SS). This specializes to the usual definition of a bounded set in a normed vector space: a set is bounded if it is contained in a ball of some finite radius r>0r \gt 0.

A linear operator A:V 1V 2A: V_1\to V_2 between topological vector spaces is bounded if it sends each bounded set in V 1V_1 to a bounded set in V 2V_2. For normed spaces, this is equivalent to saying that it sends the unit ball to a bounded set. Between finite dimensional normed spaces, every linear operator is bounded. A linear operator between any two normed linear spaces is bounded iff it is continuous.

There is also a rich theory for unbounded operators on Hilbert spaces.

Properties

Every bounded operator on a Hilbert space has a polar decomposition.

Important classes of bounded operators are the compact operators, trace-class operator?s and Hilbert-Schmidt operator?s.

References

category: analysis

Revised on October 12, 2013 21:48:44 by Toby Bartels (98.19.41.253)