Closed linear operator

In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a partially defined 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 ${\displaystyle X}$. In practice, many operators are unbounded, but it is still desirable to make them have closed graph. Hence, they cannot be defined on all of ${\displaystyle X}$. To stay useful, they are instead defined on a proper but dense subspace, which still allows approximating any vector and keeps key tools (closures, adjoints, spectral theory) available.