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In functional analysis, it is common to endow various (infinite-dimensional) vector spaces with a variety of topologies. For instance, a normed vector space can be given the strong topology as well as the weak topology; if the vector space has a predual, it also has a weak-* topology. Similarly, spaces of operators have a number of useful topologies on them, including the operator norm topology, strong operator topology, and the weak operator topology. For function spaces, one can use topologies associated to various modes of convergence, such as uniform convergence, pointwise convergence, locally uniform convergence, or convergence in the sense of distributions. (A small minority of such modes are not topologisable, though, the most common of which is pointwise almost everywhere convergence; see Exercise 8 of this previous post).

Some of these topologies are much stronger than others (in that they contain many more open sets, or equivalently that they have many fewer convergent sequences and nets). However, even the weakest topologies used in analysis (e.g. convergence in distributions) tend to be Hausdorff, since this at least ensures the uniqueness of limits of sequences and nets, which is a fundamentally useful feature for analysis. On the other hand, some Hausdorff topologies used are “better” than others in that many more analysis tools are available for those topologies. In particular, topologies that come from Banach space norms are particularly valued, as such topologies (and their attendant norm and metric structures) grant access to many convenient additional results such as the Baire category theorem, the uniform boundedness principle, the open mapping theorem, and the closed graph theorem.

Of course, most topologies placed on a vector space will not come from Banach space norms. For instance, if one takes the space ${C_0({\bf R})}$ of continuous functions on ${{\bf R}}$ that converge to zero at infinity, the topology of uniform convergence comes from a Banach space norm on this space (namely, the uniform norm ${\| \|_{L^\infty}}$), but the topology of pointwise convergence does not; and indeed all the other usual modes of convergence one could use here (e.g. ${L^1}$ convergence, locally uniform convergence, convergence in measure, etc.) do not arise from Banach space norms.

I recently realised (while teaching a graduate class in real analysis) that the closed graph theorem provides a quick explanation for why Banach space topologies are so rare:

Proposition 1 Let ${V = (V, {\mathcal F})}$ be a Hausdorff topological vector space. Then, up to equivalence of norms, there is at most one norm ${\| \|}$ one can place on ${V}$ so that ${(V,\| \|)}$ is a Banach space whose topology is at least as strong as ${{\mathcal F}}$. In particular, there is at most one topology stronger than ${{\mathcal F}}$ that comes from a Banach space norm.

Proof: Suppose one had two norms ${\| \|_1, \| \|_2}$ on ${V}$ such that ${(V, \| \|_1)}$ and ${(V, \| \|_2)}$ were both Banach spaces with topologies stronger than ${{\mathcal F}}$. Now consider the graph of the identity function ${\hbox{id}: V \rightarrow V}$ from the Banach space ${(V, \| \|_1)}$ to the Banach space ${(V, \| \|_2)}$. This graph is closed; indeed, if ${(x_n,x_n)}$ is a sequence in this graph that converged in the product topology to ${(x,y)}$, then ${x_n}$ converges to ${x}$ in ${\| \|_1}$ norm and hence in ${{\mathcal F}}$, and similarly ${x_n}$ converges to ${y}$ in ${\| \|_2}$ norm and hence in ${{\mathcal F}}$. But limits are unique in the Hausdorff topology ${{\mathcal F}}$, so ${x=y}$. Applying the closed graph theorem (see also previous discussions on this theorem), we see that the identity map is continuous from ${(V, \| \|_1)}$ to ${(V, \| \|_2)}$; similarly for the inverse. Thus the norms ${\| \|_1, \| \|_2}$ are equivalent as claimed. $\Box$

By using various generalisations of the closed graph theorem, one can generalise the above proposition to Fréchet spaces, or even to F-spaces. The proposition can fail if one drops the requirement that the norms be stronger than a specified Hausdorff topology; indeed, if ${V}$ is infinite dimensional, one can use a Hamel basis of ${V}$ to construct a linear bijection on ${V}$ that is unbounded with respect to a given Banach space norm ${\| \|}$, and which can then be used to give an inequivalent Banach space structure on ${V}$.

One can interpret Proposition 1 as follows: once one equips a vector space with some “weak” (but still Hausdorff) topology, there is a canonical choice of “strong” topology one can place on that space that is stronger than the “weak” topology but arises from a Banach space structure (or at least a Fréchet or F-space structure), provided that at least one such structure exists. In the case of function spaces, one can usually use the topology of convergence in distribution as the “weak” Hausdorff topology for this purpose, since this topology is weaker than almost all of the other topologies used in analysis. This helps justify the common practice of describing a Banach or Fréchet function space just by giving the set of functions that belong to that space (e.g. ${{\mathcal S}({\bf R}^n)}$ is the space of Schwartz functions on ${{\bf R}^n}$) without bothering to specify the precise topology to serve as the “strong” topology, since it is usually understood that one is using the canonical such topology (e.g. the Fréchet space structure on ${{\mathcal S}({\bf R}^n)}$ given by the usual Schwartz space seminorms).

Of course, there are still some topological vector spaces which have no “strong topology” arising from a Banach space at all. Consider for instance the space ${c_c({\bf N})}$ of finitely supported sequences. A weak, but still Hausdorff, topology to place on this space is the topology of pointwise convergence. But there is no norm ${\| \|}$ stronger than this topology that makes this space a Banach space. For, if there were, then letting ${e_1,e_2,e_3,\dots}$ be the standard basis of ${c_c({\bf N})}$, the series ${\sum_{n=1}^\infty 2^{-n} e_n / \| e_n \|}$ would have to converge in ${\| \|}$, and hence pointwise, to an element of ${c_c({\bf N})}$, but the only available pointwise limit for this series lies outside of ${c_c({\bf N})}$. But I do not know if there is an easily checkable criterion to test whether a given vector space (equipped with a Hausdorff “weak” toplogy) can be equipped with a stronger Banach space (or Fréchet space or ${F}$-space) topology.

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.

$\Box$

The notion of what it means for a subset E of a space X to be “small” varies from context to context.  For instance, in measure theory, when $X = (X, {\mathcal X}, \mu)$ is a measure space, one useful notion of a “small” set is that of a null set: a set E of measure zero (or at least contained in a set of measure zero).  By countable additivity, countable unions of null sets are null.  Taking contrapositives, we obtain

Lemma 1. (Pigeonhole principle for measure spaces) Let $E_1, E_2, \ldots$ be an at most countable sequence of measurable subsets of a measure space X.  If $\bigcup_n E_n$ has positive measure, then at least one of the $E_n$ has positive measure.

Now suppose that X was a Euclidean space ${\Bbb R}^d$ with Lebesgue measure m.  The Lebesgue differentiation theorem easily implies that having positive measure is equivalent to being “dense” in certain balls:

Proposition 1. Let $E$ be a measurable subset of ${\Bbb R}^d$.  Then the following are equivalent:

1. E has positive measure.
2. For any $\varepsilon > 0$, there exists a ball B such that $m( E \cap B ) \geq (1-\varepsilon) m(B)$.

Thus one can think of a null set as a set which is “nowhere dense” in some measure-theoretic sense.

It turns out that there are analogues of these results when the measure space $X = (X, {\mathcal X}, \mu)$  is replaced instead by a complete metric space $X = (X,d)$.  Here, the appropriate notion of a “small” set is not a null set, but rather that of a nowhere dense set: a set E which is not dense in any ball, or equivalently a set whose closure has empty interior.  (A good example of a nowhere dense set would be a proper subspace, or smooth submanifold, of ${\Bbb R}^d$, or a Cantor set; on the other hand, the rationals are a dense subset of ${\Bbb R}$ and thus clearly not nowhere dense.)   We then have the following important result:

Theorem 1. (Baire category theorem). Let $E_1, E_2, \ldots$ be an at most countable sequence of subsets of a complete metric space X.  If $\bigcup_n E_n$ contains a ball B, then at least one of the $E_n$ is dense in a sub-ball B’ of B (and in particular is not nowhere dense).  To put it in the contrapositive: the countable union of nowhere dense sets cannot contain a ball.

Exercise 1. Show that the Baire category theorem is equivalent to the claim that in a complete metric space, the countable intersection of open dense sets remain dense.  $\diamond$

Exercise 2. Using the Baire category theorem, show that any non-empty complete metric space without isolated points is uncountable.  (In particular, this shows that Baire category theorem can fail for incomplete metric spaces such as the rationals ${\Bbb Q}$.)  $\diamond$

To quickly illustrate an application of the Baire category theorem, observe that it implies that one cannot cover a finite-dimensional real or complex vector space ${\Bbb R}^n, {\Bbb C}^n$ by a countable number of proper subspaces.  One can of course also establish this fact by using Lebesgue measure on this space.  However, the advantage of the Baire category approach is that it also works well in infinite dimensional complete normed vector spaces, i.e. Banach spaces, whereas the measure-theoretic approach runs into significant difficulties in infinite dimensions.  This leads to three fundamental equivalences between the qualitative theory of continuous linear operators on Banach spaces (e.g. finiteness, surjectivity, etc.) to the quantitative theory (i.e. estimates):

1. The uniform boundedness principle, that equates the qualitative boundedness (or convergence) of a family of continuous operators with their quantitative boundedness.
2. The open mapping theorem, that equates the qualitative solvability of a linear problem Lu = f with the quantitative solvability.
3. The closed graph theorem, that equates the qualitative regularity of a (weakly continuous) operator T with the quantitative regularity of that operator.

Strictly speaking, these theorems are not used much directly in practice, because one usually works in the reverse direction (i.e. first proving quantitative bounds, and then deriving qualitative corollaries); but the above three theorems help explain why we usually approach qualitative problems in functional analysis via their quantitative counterparts.