Closed graph theorem (functional analysis)

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a result connecting the continuity of certain kinds of functions to a topological property of their graph. In its most elementary form, the closed graph theorem states that a linear function between two Banach spaces is continuous if and only if the graph of that function is closed. The closed graph theorem has extensive application throughout functional analysis, because it can control whether a partially-defined linear operator admits continuous extensions. For this reason, it has been generalized to many circumstances beyond the elementary formulation above. The closed graph theorem is a result about linear map between two vector spaces endowed with topologies making them into topological vector spaces (TVSs). We will henceforth assume that and are topological vector spaces, such as Banach spaces for example, and that Cartesian products, such as are endowed with the product topology. The of this function is the subset of where denotes the function's domain. The map is said to have a (in ) if its graph is a closed subset of product space (with the usual product topology). Similarly, is said to have a if is a sequentially closed subset of A is a linear map whose graph is closed (it need not be continuous or bounded). It is common in functional analysis to call such maps "closed", but this should not be confused the non-equivalent notion of a "closed map" that appears in general topology. Partial functions It is common in functional analysis to consider partial functions, which are functions defined on a dense subset of some space A partial function is declared with the notation which indicates that has prototype (that is, its domain is and its codomain is ) and that is a dense subset of Since the domain is denoted by it is not always necessary to assign a symbol (such as ) to a partial function's domain, in which case the notation or may be used to indicate that is a partial function with codomain whose domain is a dense subset of A densely defined linear operator between vector spaces is a partial function whose domain is a dense vector subspace of a TVS such that is a linear map.