Contents

# Contents

## 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 $\lambda U = \{\lambda x | x\in U\}$ 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 $r \gt 0$.

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.

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

## Properties

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

The Gelfand-Naimark theorem says that every C*-algebra is isomorphic to a $C^\ast$-algebra of bounded linear operators on a Hilbert space.

## Examples

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

category: analysis