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Closed graph theorem (functional analysis)
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<p>In mathematics, particularly in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the closed graph theorem is a result connecting the <a href="/facts/Continuous_function/sKbl02pB">continuity</a> of a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> to a topological property of their <a href="/facts/Graph_(function)/htYX0BGX">graph</a>. Precisely, the theorem states that a linear operator between two <a href="/facts/Banach_spaces/sDEIk7EC">Banach spaces</a> is continuous <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the graph of the operator is closed (such an operator is called a <a href="/facts/Closed_linear_operator/l01mnNoJ">closed linear operator</a>; see also <a href="/facts/Closed_graph_property/canolygI">closed graph property</a>). </p><p>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. </p>