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.
A blog post by T. Tao lists several closed graph theorems throughout mathematics.

Graphs and maps with closed graphs

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 .
Any continuous function into a Hausdorff space has a closed graph
Any linear map, between two topological vector spaces whose topologies are complete with respect to translation invariant metrics, and if in addition is sequentially continuous in the sense of the product topology, then the map is continuous and its graph,, is necessarily closed. Conversely, if is such a linear map with, in place of, the graph of is known to be closed in the Cartesian product space, then is continuous and therefore necessarily sequentially continuous.

Examples of continuous maps that do ''not'' have a closed graph

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. Let be defined by and for all. Then is continuous but its graph is closed in.

Closed graph theorem in point-set topology

In point-set topology, the closed graph theorem states the following:
If X, Y are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see.
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.
Also, closed linear operators in functional analysis are typically not continuous.

In functional analysis

If is a linear operator between topological vector spaces 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.
The original result has been generalized many times.
A well known version of the closed graph theorems is the following.
The theorem is a consequence of the open [mapping theorem (functional analysis)|open mapping theorem]; see below.

Relation to the open mapping theorem

Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way. Let be any map. Then it factors as
Now, is the inverse of the projection. So, if the open mapping theorem holds for ; i.e., is an open mapping, then is continuous and then is continuous.
For example, the above argument applies if is a linear operator between Banach spaces with closed graph, or if is a map with closed graph between compact Hausdorff spaces.