We give the more general definition of when a Y-valued function or set-valued function defined on a subset S of X has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace S of a topological vector space X (and not necessarily defined on all of X).
When reading literature in functional analysis, if f : X → Y is a linear map between topological vector spaces (TVSs) (e.g. Banach spaces) then "f is closed" will almost always means the following: Otherwise, especially in literature about point-set topology, "f is closed" may instead mean the following: These two definitions of "closed map" are not equivalent.
If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.
If f : X → Y is a function then the following are equivalent: and if Y is a Hausdorff space that is compact, then we may add to this list:
is a function then it is said to have a closed graph if it satisfies any of the following are equivalent conditions: and if
are first-countable spaces then we may add to this list: Function with a sequentially closed graph If
Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous.
For examples in functional analysis, see continuous linear operator.