Let and be two Banach spaces. A compact operator (completely continuous operator) is a linear operator that maps the unit ball in to a set in with compact closure. It can be shown that a compact operator is necessarily a bounded operator.
The set of all compact operators on , commonly denoted by , is a closed two-sided ideal of the set of all bounded operators on , .
Any bounded operator which is the norm limit of a sequence of finite rank operators is compact. In the case of Hilbert spaces, the converse is also true. That is, any compact operator on a Hilbert space is a norm limit of finite rank operators.
Example 1 (Integral operators)
Let , with , be a continuous function. The operator defined by
|Date of creation||2013-03-22 14:26:59|
|Last modified on||2013-03-22 14:26:59|
|Last modified by||mhale (572)|