Closed linear operator

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator.

The closed graph theorem says a linear operator ${\displaystyle f:X\to Y}$ between Banach spaces is a closed operator if and only if it is a bounded operator and the domain of the operator is ${\displaystyle X}$. Hence, a closed linear operator that is used in practice is typically only defined on a dense subspace of a Banach space.