Given a function {f: X \rightarrow Y} between two sets {X, Y}, we can form the graph

\displaystyle  \Sigma := \{ (x,f(x)): x\in X \},

which is a subset of the Cartesian product {X \times Y}.

There are a number of “closed graph theorems” in mathematics which relate the regularity properties of the function {f} with the closure properties of the graph {\Sigma}, assuming some “completeness” properties of the domain {X} and range {Y}. The most famous of these is the closed graph theorem from functional analysis, which I phrase as follows:

Theorem 1 (Closed graph theorem (functional analysis)) Let {X, Y} be complete normed vector spaces over the reals (i.e. Banach spaces). Then a function {f: X \rightarrow Y} is a continuous linear transformation if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is both linearly closed (i.e. it is a linear subspace of {X \times Y}) and topologically closed (i.e. closed in the product topology of {X \times Y}).

I like to think of this theorem as linking together qualitative and quantitative notions of regularity preservation properties of an operator {f}; see this blog post for further discussion.

The theorem is equivalent to the assertion that any continuous linear bijection {f: X \rightarrow Y} from one Banach space to another is necessarily an isomorphism in the sense that the inverse map is also continuous and linear. Indeed, to see that this claim implies the closed graph theorem, one applies it to the projection from {\Sigma} to {X}, which is a continuous linear bijection; conversely, to deduce this claim from the closed graph theorem, observe that the graph of the inverse {f^{-1}} is the reflection of the graph of {f}. As such, the closed graph theorem is a corollary of the open mapping theorem, which asserts that any continuous linear surjection from one Banach space to another is open. (Conversely, one can deduce the open mapping theorem from the closed graph theorem by quotienting out the kernel of the continuous surjection to get a bijection.)

It turns out that there is a closed graph theorem (or equivalent reformulations of that theorem, such as an assertion that bijective morphisms between sufficiently “complete” objects are necessarily isomorphisms, or as an open mapping theorem) in many other categories in mathematics as well. Here are some easy ones:

Theorem 2 (Closed graph theorem (linear algebra)) Let {X, Y} be vector spaces over a field {k}. Then a function {f: X \rightarrow Y} is a linear transformation if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is linearly closed.

Theorem 3 (Closed graph theorem (group theory)) Let {X, Y} be groups. Then a function {f: X \rightarrow Y} is a group homomorphism if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is closed under the group operations (i.e. it is a subgroup of {X \times Y}).

Theorem 4 (Closed graph theorem (order theory)) Let {X, Y} be totally ordered sets. Then a function {f: X \rightarrow Y} is monotone increasing if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is totally ordered (using the product order on {X \times Y}).

Remark 1 Similar results to the above three theorems (with similarly easy proofs) hold for other algebraic structures, such as rings (using the usual product of rings), modules, algebras, or Lie algebras, groupoids, or even categories (a map between categories is a functor iff its graph is again a category). (ADDED IN VIEW OF COMMENTS: further examples include affine spaces and {G}-sets (sets with an action of a given group {G}).) There are also various approximate versions of this theorem that are useful in arithmetic combinatorics, that relate the property of a map {f} being an “approximate homomorphism” in some sense with its graph being an “approximate group” in some sense. This is particularly useful for this subfield of mathematics because there are currently more theorems about approximate groups than about approximate homomorphisms, so that one can profitably use closed graph theorems to transfer results about the former to results about the latter.

A slightly more sophisticated result in the same vein:

Theorem 5 (Closed graph theorem (point set topology)) Let {X, Y} be compact Hausdorff spaces. Then a function {f: X \rightarrow Y} is continuous if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is topologically closed.

Indeed, the “only if” direction is easy, while for the “if” direction, note that if {\Sigma} is a closed subset of {X \times Y}, then it is compact Hausdorff, and the projection map from {\Sigma} to {X} is then a bijective continuous map between compact Hausdorff spaces, which is then closed, thus open, and hence a homeomorphism, giving the claim.

Note that the compactness hypothesis is necessary: for instance, the function {f: {\bf R} \rightarrow {\bf R}} defined by {f(x) := 1/x} for {x \neq 0} and {f(0)=0} for {x=0} is a function which has a closed graph, but is discontinuous.

A similar result (but relying on a much deeper theorem) is available in algebraic geometry, as I learned after asking this MathOverflow question:

Theorem 6 (Closed graph theorem (algebraic geometry)) Let {X, Y} be normal projective varieties over an algebraically closed field {k} of characteristic zero. Then a function {f: X \rightarrow Y} is a regular map if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is Zariski-closed.

Proof: (Sketch) For the only if direction, note that the map {x \mapsto (x,f(x))} is a regular map from the projective variety {X} to the projective variety {X \times Y} and is thus a projective morphism, hence is proper. In particular, the image {\Sigma} of {X} under this map is Zariski-closed.

Conversely, if {\Sigma} is Zariski-closed, then it is also a projective variety, and the projection {(x,y) \mapsto x} is a projective morphism from {\Sigma} to {X}, which is clearly quasi-finite; by the characteristic zero hypothesis, it is also separated. Applying (Grothendieck’s form of) Zariski’s main theorem, this projection is the composition of an open immersion and a finite map. As projective varieties are complete, the open immersion is an isomorphism, and so the projection from {\Sigma} to {X} is finite. Being injective and separable, the degree of this finite map must be one, and hence {k(\Sigma)} and {k(X)} are isomorphic, hence (by normality of {X}) {k[\Sigma]} is contained in (the image of) {k[X]}, which makes the map from {X} to {\Sigma} regular, which makes {f} regular. \Box

The counterexample of the map {f: k \rightarrow k} given by {f(x) := 1/x} for {x \neq 0} and {f(0) := 0} demonstrates why the projective hypothesis is necessary. The necessity of the normality condition (or more precisely, a weak normality condition) is demonstrated by (the projective version of) the map {(t^2,t^3) \mapsto t} from the cusipdal curve {\{ (t^2,t^3): t \in k \}} to {k}. (If one restricts attention to smooth varieties, though, normality becomes automatic.) The necessity of characteristic zero is demonstrated by (the projective version of) the inverse of the Frobenius map {x \mapsto x^p} on a field {k} of characteristic {p}.

There are also a number of closed graph theorems for topological groups, of which the following is typical (see Exercise 3 of these previous blog notes):

Theorem 7 (Closed graph theorem (topological group theory)) Let {X, Y} be {\sigma}-compact, locally compact Hausdorff groups. Then a function {X \rightarrow Y} is a continuous homomorphism if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is both group-theoretically closed and topologically closed.

The hypotheses of being {\sigma}-compact, locally compact, and Hausdorff can be relaxed somewhat, but I doubt that they can be eliminated entirely (though I do not have a ready counterexample for this).

In several complex variables, it is a classical theorem (see e.g. Lemma 4 of this blog post) that a holomorphic function from a domain in {{\bf C}^n} to {{\bf C}^n} is locally injective if and only if it is a local diffeomorphism (i.e. its derivative is everywhere non-singular). This leads to a closed graph theorem for complex manifolds:

Theorem 8 (Closed graph theorem (complex manifolds)) Let {X, Y} be complex manifolds. Then a function {f: X \rightarrow Y} is holomorphic if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is a complex manifold (using the complex structure inherited from {X \times Y}) of the same dimension as {X}.

Indeed, one applies the previous observation to the projection from {\Sigma} to {X}. The dimension requirement is needed, as can be seen from the example of the map {f: {\bf C} \rightarrow {\bf C}} defined by {f(z) =1/z} for {z \neq 0} and {f(0)=0}.

(ADDED LATER:) There is a real analogue to the above theorem:

Theorem 9 (Closed graph theorem (real manifolds)) Let {X, Y} be real manifolds. Then a function {f: X \rightarrow Y} is continuous if and only if the graph {\Sigma := \{ (x,f(x)): x \in X \}} is a real manifold of the same dimension as {X}.

This theorem can be proven by applying invariance of domain (discussed in this previous post) to the projection of {\Sigma} to {X}, to show that it is open if {\Sigma} has the same dimension as {X}.

Note though that the analogous claim for smooth real manifolds fails: the function {f: {\bf R} \rightarrow {\bf R}} defined by {f(x) := x^{1/3}} has a smooth graph, but is not itself smooth.

(ADDED YET LATER:) Here is an easy closed graph theorem in the symplectic category:

Theorem 10 (Closed graph theorem (symplectic geometry)) Let {X = (X,\omega_X)} and {Y = (Y,\omega_Y)} be smooth symplectic manifolds of the same dimension. Then a smooth map {f: X \rightarrow Y} is a symplectic morphism (i.e. {f^* \omega_Y = \omega_X}) if and only if the graph {\Sigma := \{(x,f(x)): x \in X \}} is a Lagrangian submanifold of {X \times Y} with the symplectic form {\omega_X \oplus -\omega_Y}.

In view of the symplectic rigidity phenomenon, it is likely that the smoothness hypotheses on {f,X,Y} can be relaxed substantially, but I will not try to formulate such a result here.

There are presumably many further examples of closed graph theorems (or closely related theorems, such as criteria for inverting a morphism, or open mapping type theorems) throughout mathematics; I would be interested to know of further examples.