Closed graph theorem (functional analysis)

In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed (such an operator is called a closed linear operator; see also closed graph property).

An important question in functional analysis is whether a given linear operator is continuous (or bounded). The closed graph theorem gives one answer to that question.