When Y is a Hilbert space , it is true that any compact operator is a limit of finite - rank operators .
The class of compact operators can be defined alternatively as the closure of the set of finite - rank operators in the norm topology .
When Y is a Hilbert space , it is true that any compact operator is a limit of finite - rank operators , so that the class of compact operators can be defined alternatively as the closure of the set of finite - rank operators in the norm topology .