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[1] by T. Tao lists several closed graph theorems throughout mathematics.

is a map between topological spaces then the graph of

Any continuous function into a Hausdorff space has a closed graph (see § Closed graph theorem in point-set topology) Any linear map,

between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a)

is sequentially continuous in the sense of the product topology, then the map

is continuous and its graph, Gr L, is necessarily closed.

is such a linear map with, in place of (1a), the graph of

is (1b) known to be closed in the Cartesian product space

is continuous but its graph, which is the diagonal

is continuous but does not have a closed graph.

is continuous but its graph is not closed in

[4] In point-set topology, the closed graph theorem states the following: Closed graph theorem[5] — If

(Note that compactness and Hausdorffness do not imply each other.)

is compact, we can take a finite open covering of

, a contradiction since it is supposed to be disjoint from the graph of

If X, Y are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see § Relation to the open mapping theorem.

is the real line, which allows the discontinuous function with closed graph

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

Closed graph theorem for set-valued functions[6] — For a Hausdorff compact range space

is a linear operator between topological vector spaces (TVSs) then we say that

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.

Theorem[7][8] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.

Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way.

So, if the open mapping theorem holds for

is a linear operator between Banach spaces with closed graph, or if

is a map with closed graph between compact Hausdorff spaces.