nLab
bounded operator

Idea

In a (Hausdorff) topological vector space one can consider bounded sets: a set $S$ is bounded if it is absorbed by any open neighborhood $U$ of zero (i.e. a dilated multiple $\lambda U=\{\lambda x\mid x\in U\}$ contains $S$). 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 $r>0$.

A linear operator $A:V_1\to V_2$ between topological vector spaces is bounded if it sends each bounded set in $V_1$ to a bounded set in $V_2$. 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.

Related concepts

• operator

• linear operator

• bounded linear operator / unbounded linear operator

• operator algebra

References

• wikipedia: bounded operator

category: analysis