A closed operator is a linear operator defined on a dense subset of a Hilbert space that is closed in the sense that its graph is a closed set in the product space. This property ensures that if a sequence converges in the domain of the operator, then the image of that sequence under the operator also converges. Closed operators are significant in functional analysis as they extend the concept of bounded operators and are essential for the spectral theory and the study of unbounded operators.
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