In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphss. 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

is a map between topological spaces then the graph It is said that is closed is a closed subset (with the product topology).

Any continuous function into a Hausdorff space has a closed graph (see )

is continuous and its graph, is continuous and therefore necessarily sequentially continuous.

Examples of continuous maps that do not have a closed graph

is Hausdorff. is continuous but does have a closed graph.

with the usual Euclidean topology with the indiscrete topology is is continuous but its graph is .

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 .

.

Also, closed linear operators in functional analysis (linear operators with closed graphs) are typically not continuous.

For set-valued functions

In functional analysis

is a linear operator between topological vector spaces is a closed operator 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; see below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).

Relation to the open mapping theorem

Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way. be any map. Then it factors as

1. .

is continuous (as the composition of continuous maps).

is a map with closed graph between compact Hausdorff spaces.

See also

Notes

References

Bibliography