Closed linear operator

Article summary

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 states that a linear operator f : X \to Y between Banach spaces with full domain X is a closed operator if and only if it is a bounded operator. In practice, many operators are unbounded, but it is still desirable to make them have a closed graph. Hence, they cannot be defined on all of 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.

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