Closed graph theorem

In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. Closed graph If is a map between topological spaces then the graph of is the set or equivalently, It is said that the graph of is closed if is a closed subset of (with the product topology). Any continuous function into a Hausdorff space has a closed graph. Any linear map, between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) is sequentially continuous in the sense of the product topology, then the map is continuous and its graph, Gr L, is necessarily closed. Conversely, if is such a linear map with, in place of (1a), the graph of is (1b) known to be closed in the Cartesian product space , then is continuous and therefore necessarily sequentially continuous. If is any space then the identity map is continuous but its graph, which is the diagonal , is closed in if and only if is Hausdorff. In particular, if is not Hausdorff then is continuous but does have a closed graph. Let denote the real numbers with the usual Euclidean topology and let denote with the indiscrete topology (where note that is Hausdorff and that every function valued in is continuous). Let be defined by and for all . Then is continuous but its graph is closed in . In point-set topology, the closed graph theorem states the following: Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact is the real line, which allows the discontinuous function with closed graph . Closed graph theorem (functional analysis) If is a linear operator between topological vector spaces (TVSs) then we say that is a closed operator if the graph of is closed in when is endowed with the product topology. The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions.