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Given a function between two sets , we can form the graph
which is a subset of the Cartesian product .
There are a number of “closed graph theorems” in mathematics which relate the regularity properties of the function with the closure properties of the graph , assuming some “completeness” properties of the domain and range . 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 be complete normed vector spaces over the reals (i.e. Banach spaces). Then a function is a continuous linear transformation if and only if the graph is both linearly closed (i.e. it is a linear subspace of ) and topologically closed (i.e. closed in the product topology of ).
I like to think of this theorem as linking together qualitative and quantitative notions of regularity preservation properties of an operator ; see this blog post for further discussion.
The theorem is equivalent to the assertion that any continuous linear bijection 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 to , which is a continuous linear bijection; conversely, to deduce this claim from the closed graph theorem, observe that the graph of the inverse is the reflection of the graph of . 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 be vector spaces over a field . Then a function is a linear transformation if and only if the graph is linearly closed.
Theorem 3 (Closed graph theorem (group theory)) Let be groups. Then a function is a group homomorphism if and only if the graph is closed under the group operations (i.e. it is a subgroup of ).
Theorem 4 (Closed graph theorem (order theory)) Let be totally ordered sets. Then a function is monotone increasing if and only if the graph is totally ordered (using the product order on ).
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 -sets (sets with an action of a given group ).) There are also various approximate versions of this theorem that are useful in arithmetic combinatorics, that relate the property of a map 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 be compact Hausdorff spaces. Then a function is continuous if and only if the graph is topologically closed.
Indeed, the “only if” direction is easy, while for the “if” direction, note that if is a closed subset of , then it is compact Hausdorff, and the projection map from to 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 defined by for and for 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 be normal projective varieties over an algebraically closed field of characteristic zero. Then a function is a regular map if and only if the graph is Zariski-closed.
Proof: (Sketch) For the only if direction, note that the map is a regular map from the projective variety to the projective variety and is thus a projective morphism, hence is proper. In particular, the image of under this map is Zariski-closed.
Conversely, if is Zariski-closed, then it is also a projective variety, and the projection is a projective morphism from to , 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 to is finite. Being injective and separable, the degree of this finite map must be one, and hence and are isomorphic, hence (by normality of ) is contained in (the image of) , which makes the map from to regular, which makes regular.
The counterexample of the map given by for and 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 from the cusipdal curve to . (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 on a field of characteristic .
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 be -compact, locally compact Hausdorff groups. Then a function is a continuous homomorphism if and only if the graph is both group-theoretically closed and topologically closed.
The hypotheses of being -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 to 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 be complex manifolds. Then a function is holomorphic if and only if the graph is a complex manifold (using the complex structure inherited from ) of the same dimension as .
Indeed, one applies the previous observation to the projection from to . The dimension requirement is needed, as can be seen from the example of the map defined by for and .
(ADDED LATER:) There is a real analogue to the above theorem:
Theorem 9 (Closed graph theorem (real manifolds)) Let be real manifolds. Then a function is continuous if and only if the graph is a real manifold of the same dimension as .
Note though that the analogous claim for smooth real manifolds fails: the function defined by 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 and be smooth symplectic manifolds of the same dimension. Then a smooth map is a symplectic morphism (i.e. ) if and only if the graph is a Lagrangian submanifold of with the symplectic form .
In view of the symplectic rigidity phenomenon, it is likely that the smoothness hypotheses on 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.
In his wonderful article “On proof and progress in mathematics“, Bill Thurston describes (among many other topics) how one’s understanding of given concept in mathematics (such as that of the derivative) can be vastly enriched by viewing it simultaneously from many subtly different perspectives; in the case of the derivative, he gives seven standard such perspectives (infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic) and then mentions a much later perspective in the sequence (as describing a flat connection for a graph).
One can of course do something similar for many other fundamental notions in mathematics. For instance, the notion of a group can be thought of in a number of (closely related) ways, such as the following:
- (0) Motivating examples: A group is an abstraction of the operations of addition/subtraction or multiplication/division in arithmetic or linear algebra, or of composition/inversion of transformations.
- (1) Universal algebraic: A group is a set with an identity element , a unary inverse operation , and a binary multiplication operation obeying the relations (or axioms) , , for all .
- (2) Symmetric: A group is all the ways in which one can transform a space to itself while preserving some object or structure on this space.
- (3) Representation theoretic: A group is identifiable with a collection of transformations on a space which is closed under composition and inverse, and contains the identity transformation.
- (4) Presentation theoretic: A group can be generated by a collection of generators subject to some number of relations.
- (5) Topological: A group is the fundamental group of a connected topological space .
- (6) Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system.
- (7) Category theoretic: A group is a category with one object, in which all morphisms have inverses.
- (8) Quantum: A group is the classical limit of a quantum group.
One can view a large part of group theory (and related subjects, such as representation theory) as exploring the interconnections between various of these perspectives. As one’s understanding of the subject matures, many of these formerly distinct perspectives slowly merge into a single unified perspective.
From a recent talk by Ezra Getzler, I learned a more sophisticated perspective on a group, somewhat analogous to Thurston’s example of a sophisticated perspective on a derivative (and coincidentally, flat connections play a central role in both):
- (37) Sheaf theoretic: A group is identifiable with a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of -simplices are bijective for (and merely surjective for ).
This interpretation of the group concept is apparently due to Grothendieck, though it is motivated also by homotopy theory. One of the key advantages of this interpretation is that it generalises easily to the notion of an -group (simply by replacing with in (37)), whereas the other interpretations listed earlier require a certain amount of subtlety in order to generalise correctly (in particular, they usually themselves require higher-order notions, such as -categories).
The connection of (37) with any of the other perspectives of a group is elementary, but not immediately obvious; I enjoyed working out exactly what the connection was, and thought it might be of interest to some readers here, so I reproduce it below the fold.
[Note: my reconstruction of Grothendieck's perspective, and of the appropriate terminology, is likely to be somewhat inaccurate in places: corrections are of course very welcome.]
I’ve just uploaded to the arXiv my joint paper with Tim Austin, “On the testability and repair of hereditary hypergraph properties“, which has been submitted to Random Structures and Algorithms. In this paper we prove some positive and negative results for the testability (and the local repairability) of various properties of directed or undirected graphs and hypergraphs, which can be either monochromatic or multicoloured.
The negative results have already been discussed in a previous posting of mine, so today I will focus on the positive results. The property testing results here are finitary results, but it turns out to be rather convenient to use a certain correspondence principle (the hypergraph version of the Furstenberg correspondence principle) to convert the question into one about exchangeable probability measures on spaces of hypergraphs (i.e. on random hypergraphs whose probability distribution is invariant under exchange of vertices). Such objects are also closely related to the”graphons” and “hypergraphons” that emerge as graph limits, as studied by Lovasz-Szegedy, Elek-Szegedy, and others. Somewhat amusingly, once one does so, it then becomes convenient to keep track of objects indexed by vertex sets and how they are exchanged via the language of category theory, and in particular using the concept of a natural transformation to describe such objects as exchangeable measures, graph colourings, and local modification rules. I will try to sketch out some of these connections, after describing the main positive results.
Before we begin or study of dynamical systems, topological dynamical systems, and measure-preserving systems (as defined in the previous lecture), it is convenient to give these three classes the structure of a category. One of the basic insights of category theory is that a mathematical objects in a given class (such as dynamical systems) are best studied not in isolation, but in relation to each other, via morphisms. Furthermore, many other basic concepts pertaining to these objects (e.g. subobjects, factors, direct sums, irreducibility, etc.) can be defined in terms of these morphisms. One advantage of taking this perspective here is that it provides a unified way of defining these concepts for the three different categories of dynamical systems, topological dynamical systems, and measure-preserving systems that we will study in this course, thus sparing us the need to give any of our definitions (except for our first one below) in triplicate.