Editing Closed graph theorem (section)

== In functional analysis ==
{{Main|Closed graph theorem (functional analysis)}}

If <math>T : X \to Y</math> is a linear operator between [[topological vector space]]s (TVSs) then we say that <math>T</math> is a '''[[closed linear operator|closed operator]]''' if the graph of <math>T</math> is closed in <math>X \times Y</math> when <math>X \times Y</math> 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. 
The original result has been generalized many times. 
A well known version of the closed graph theorems is the following.

{{Math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|p=78}}<ref>{{harvtxt|Trèves|2006}}, p. 173</ref>|math_statement=
A linear map between two [[F-space]]s (e.g. [[Banach space]]s) is continuous if and only if its graph is closed.
}}

The theorem is a consequence of the [[open mapping theorem (functional analysis)|open mapping theorem]]; see {{section link|| Relation to the open mapping theorem}} below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).