bounded operator


Operator algebra

Functional analysis




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.

The quotient of the bounded operators by the compact operators is called the Calkin algebra.


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

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


category: analysis

Revised on October 17, 2016 09:13:58 by Urs Schreiber (