Closed graph theorem

Template:Short description Template:About Template:Multiple image 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<ref name="Tao">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> by T. Tao lists several closed graph theorems throughout mathematics.

Graphs and maps with closed graphsEdit

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If <math>f : X \to Y</math> is a map between topological spaces then the graph of <math>f</math> is the set <math>\Gamma_f := \{ (x, f(x)) : x \in X \}</math> or equivalently, <math display=block>\Gamma_f := \{ (x, y) \in X \times Y : y = f(x) \}</math> It is said that the graph of <math>f</math> is closed if <math>\Gamma_f</math> is a closed subset of <math>X \times Y</math> (with the product topology).

Any continuous function into a Hausdorff space has a closed graph (see Template:Section link)

Any linear map, <math>L : X \to Y,</math> between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) <math>L</math> is sequentially continuous in the sense of the product topology, then the map <math>L</math> is continuous and its graph, Template:Math, is necessarily closed. Conversely, if <math>L</math> is such a linear map with, in place of (1a), the graph of <math>L</math> is (1b) known to be closed in the Cartesian product space <math>X \times Y</math>, then <math>L</math> is continuous and therefore necessarily sequentially continuous.Template:Sfn

Examples of continuous maps that do not have a closed graphEdit

If <math>X</math> is any space then the identity map <math>\operatorname{Id} : X \to X</math> is continuous but its graph, which is the diagonal <math>\Gamma_{\operatorname{Id}} := \{ (x, x) : x \in X \},</math>, is closed in <math>X \times X</math> if and only if <math>X</math> is Hausdorff.Template:Sfn In particular, if <math>X</math> is not Hausdorff then <math>\operatorname{Id} : X \to X</math> is continuous but does Template:Em have a closed graph.

Let <math>X</math> denote the real numbers <math>\R</math> with the usual Euclidean topology and let <math>Y</math> denote <math>\R</math> with the indiscrete topology (where note that <math>Y</math> is Template:Em Hausdorff and that every function valued in <math>Y</math> is continuous). Let <math>f : X \to Y</math> be defined by <math>f(0) = 1</math> and <math>f(x) = 0</math> for all <math>x \neq 0</math>. Then <math>f : X \to Y</math> is continuous but its graph is Template:Em closed in <math>X \times Y</math>.Template:Sfn

Closed graph theorem in point-set topologyEdit

In point-set topology, the closed graph theorem states the following:

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If X, Y are compact Hausdorff spaces, then the theorem can also be deduced from the open mapping theorem for such spaces; see Template:Section link.

Non-Hausdorff spaces are rarely seen, but non-compact spaces are common. An example of non-compact <math>Y</math> is the real line, which allows the discontinuous function with closed graph <math>f(x) = \begin{cases} \frac 1 x \text{ if }x\neq 0,\\ 0\text{ else} \end{cases}</math>.

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

For set-valued functionsEdit

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In functional analysisEdit

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If <math>T : X \to Y</math> is a linear operator between topological vector spaces (TVSs) then we say that <math>T</math> is a 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.

Template:Math theorem

The theorem is a consequence of the open mapping theorem; see Template:Section link below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).

Relation to the open mapping theoremEdit

Often, the closed graph theorems are obtained as corollaries of the open mapping theorems in the following way.<ref name="Tao" /><ref>Template:Cite arXiv</ref> Let <math>f : X \to Y</math> be any map. Then it factors as

<math>f: X \overset{i}\to \Gamma_f \overset{q}\to Y</math>.

Now, <math>i</math> is the inverse of the projection <math>p: \Gamma_f \to X</math>. So, if the open mapping theorem holds for <math>p</math>; i.e., <math>p</math> is an open mapping, then <math>i</math> is continuous and then <math>f</math> is continuous (as the composition of continuous maps).

For example, the above argument applies if <math>f</math> is a linear operator between Banach spaces with closed graph, or if <math>f</math> is a map with closed graph between compact Hausdorff spaces.