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Closed graph theorem

The closed graph theorem is a cornerstone of functional analysis, asserting that a linear operator between two Banach spaces is continuous if and only if its graph—defined as the set of pairs (x, Tx) for x in the domain—is a closed subset of the product space equipped with the product topology.^[1] This equivalence holds under the assumptions that both spaces are complete normed vector spaces and the operator is defined on the entire domain.^[1] The theorem was first proved by Stefan Banach in his 1932 monograph Théorie des opérations linéaires, where it appeared as a key result linking the closedness of the graph to boundedness of the operator.^[2] Earlier precursors existed, such as Maurice Fréchet's 1906 work under separability assumptions and subsequent refinements by Hildebrandt, Gross, and Fréchet in 1912–1917 that removed those restrictions, but Banach's version established the general form for Banach spaces.^[2] A simplified proof using the Baire category theorem was provided by Juliusz Schauder in 1930, which Banach incorporated into his book.^[2] Logically equivalent to the open mapping theorem, the closed graph theorem simplifies proofs of continuity for linear operators by reducing them to verifying graph closedness, often via sequential limits rather than direct estimates.^[1] It relies on the uniform boundedness principle and has been extended to variants in Hilbert spaces, locally convex spaces, and other categories, with modern proofs emphasizing its ties to Baire category arguments.^[3] Applications abound in operator theory, including establishing boundedness of differential operators on dense domains and implications for spectral theory in Hilbert spaces.^[2]

Fundamental Concepts

Graph of a mapping

A topological space is a set X equipped with a collection \mathcal{T} of subsets of X, called open sets, that satisfies three axioms: the empty set and X itself are in \mathcal{T}; arbitrary unions of sets in \mathcal{T} are in \mathcal{T}; and finite intersections of sets in \mathcal{T} are in \mathcal{T}.^[4] This structure generalizes notions of continuity and closeness without relying on distances or metrics. Given two topological spaces (X, \mathcal{T}_X) and (Y, \mathcal{T}_Y), their Cartesian product X \times Y is the set of all ordered pairs (x, y) with x \in X and y \in Y, endowed with the product topology \mathcal{T}_{X \times Y}.^[5] The product topology is the coarsest topology on X \times Y that makes the natural projection maps \pi_X: X \times Y \to X and \pi_Y: X \times Y \to Y, defined by \pi_X(x, y) = x and \pi_Y(x, y) = y, continuous; its subbasis consists of sets of the form \pi_X^{-1}(U) for U \in \mathcal{T}_X and \pi_Y^{-1}(V) for V \in \mathcal{T}_Y.^[5] For a function f: X \to Y between sets X and Y, the graph G_f of f is the subset \{(x, f(x)) \mid x \in X\} \subseteq X \times Y.^[6] When X and Y are topological spaces, G_f is viewed as a subset of the product space X \times Y equipped with the product topology. Set-theoretically, the graph can be visualized as the collection of points in the plane (for real-valued functions) or higher-dimensional product space that lie directly "above" or "corresponding to" each input via the mapping, forming a hypersurface that encodes the entire function without redundancy.

Closed graphs in topological spaces

In topological spaces, the product topology on the Cartesian product X \times Y is defined as the coarsest topology making the natural projection maps \pi_X: X \times Y \to X and \pi_Y: X \times Y \to Y continuous, with a basis consisting of sets of the form U \times V where U is open in X and V is open in Y. A subset A \subseteq X \times Y is closed in this topology if its complement is open, equivalently, if whenever a net (x_\alpha, y_\alpha) in A converges to (x, y) in X \times Y, then (x, y) \in A. A topological space Y is Hausdorff if for every pair of distinct points y_1, y_2 \in Y, there exist disjoint open neighborhoods V_1 of y_1 and V_2 of y_2.^[7] The graph G_f of a mapping f: X \to Y between topological spaces is closed if G_f = \{(x, f(x)) \mid x \in X\} is a closed subset of X \times Y endowed with the product topology.^[8] A fundamental result establishes that continuity of f implies closedness of G_f under mild conditions on Y. Specifically, if f: X \to Y is continuous and Y is Hausdorff, then G_f is closed in X \times Y.^[9] To see this, consider the map (f, \mathrm{id}_Y): X \times Y \to Y \times Y given by (x, y) \mapsto (f(x), y); the graph G_f is the inverse image of the diagonal \Delta_Y = \{(y, y) \mid y \in Y\}, which is closed in the Hausdorff space Y \times Y, and inverse images of closed sets under continuous maps are closed.^[9] Equivalently, in terms of convergence, if a net (x_\alpha, y_\alpha) in G_f converges to (x, y) in X \times Y, then x_\alpha \to x and y_\alpha = f(x_\alpha) \to y, so continuity yields f(x_\alpha) \to f(x), and Hausdorffness ensures y = f(x), hence (x, y) \in G_f.^[9] The converse holds in simpler settings involving compactness. If X is compact and Y is Hausdorff, then a mapping f: X \to Y is continuous if and only if G_f is closed in X \times Y.^[8]

Examples and Counterexamples

Continuous mappings without closed graphs

In general topological spaces, a continuous mapping need not have a closed graph, as the absence of separation axioms like Hausdorffness can prevent the graph from being closed in the product space. This contrasts with the situation where the codomain is Hausdorff, in which case continuity implies a closed graph.^[10] A classic example occurs with the identity mapping on a non-Hausdorff space, such as the line with two origins. This space X is constructed as the set (\mathbb{R} \setminus \{0\}) \cup \{p, q\}, where p and q represent distinct origins, equipped with a basis consisting of open intervals in \mathbb{R} \setminus \{0\} and sets of the form (-a, 0) \cup \{p\} \cup (0, a) or (-a, 0) \cup \{q\} \cup (0, a) for a > 0. The identity map \mathrm{id}: X \to X is continuous by definition. However, its graph is the diagonal \Delta = \{(x, x) \mid x \in X\} \subseteq X \times X. Since X is not Hausdorff—the points p and q cannot be separated by disjoint open sets—the diagonal \Delta fails to be closed in the product topology. Specifically, the point (p, q) lies in the closure of \Delta but not in \Delta itself, as any basic open neighborhood of (p, q) in X \times X intersects \Delta. A space is Hausdorff if and only if its diagonal is closed in the product space.^[11] Another example involves a mapping into a space with the indiscrete topology, where the only open sets are the empty set and the whole space. Consider the constant function f: \mathbb{R} \to \mathbb{R}, where \mathbb{R} in the domain has the standard topology and \mathbb{R} in the codomain has the indiscrete topology; f(x) = 0 for all x. This function is continuous because the preimage of any open set in the codomain is either empty or all of \mathbb{R}, both open in the domain. The graph G = \{(x, 0) \mid x \in \mathbb{R}\} \subseteq \mathbb{R} \times \mathbb{R} (with the product topology) is not closed, as singletons like \{0\} are not closed in the indiscrete codomain. In the product topology, basic open sets are of the form U \times \mathbb{R} for U open in the domain \mathbb{R}; thus, any such neighborhood of a point (x_0, y_0) with y_0 \neq 0 contains points of G, placing (x_0, y_0) in the closure of G without belonging to it. In fact, G is dense in \mathbb{R} \times \mathbb{R}. These examples illustrate how topological properties, such as separation of points or the structure of closed sets, are essential for the graph of a continuous mapping to be closed; without them, limits in the product space can approach the graph from outside.

Closed-graph mappings that are discontinuous

A canonical example of a discontinuous mapping with a closed graph arises in the context of linear functionals on incomplete normed spaces. Consider the space c_{00} consisting of all complex (or real) sequences with only finitely many nonzero terms, equipped with the supremum norm \|(x_n)\|_\infty = \sup_n |x_n|. This space is incomplete as a normed linear space. Define the linear functional T: c_{00} \to \mathbb{C} (or \mathbb{R}) by T((x_n)) = \sum_n x_n. This functional is discontinuous because it is unbounded: for the sequence x^{(k)} with first k entries equal to 1 and the rest 0, \|x^{(k)}\|_\infty = 1 but |T(x^{(k)})| = k \to \infty as k \to \infty. However, the graph of T is closed: if (x^{(m)}) converges to x in c_{00} and T(x^{(m)}) \to y, then y = T(x), as the finite support ensures the sum converges appropriately under the limit.^[12] Another example is a nonlinear mapping from \mathbb{R} to \mathbb{R} with the standard topology. Define f(x) = 1/x for x \neq 0 and f(0) = 0. This function is discontinuous at x = 0, as \lim_{x \to 0} f(x) does not exist in \mathbb{R}. Nonetheless, the graph of f is closed in \mathbb{R} \times \mathbb{R}: it is the disjoint union of the closed set \{0\} \times \{0\} and the graphs over (-\infty, 0) and (0, \infty), each of which is closed as the preimage under the continuous projection avoids limit points at 0 due to the unbounded behavior near 0.^[12] In a topological setting without completeness, consider the domain as the rational numbers \mathbb{Q} with the subspace topology induced from \mathbb{R}, and codomain \mathbb{R} with the standard topology. Enumerate \mathbb{Q} = \{q_n : n = 1, 2, \dots \} and define f: \mathbb{Q} \to \mathbb{R} by f(q_n) = n \pi / 2. This function is discontinuous at every point in \mathbb{Q}, as it is unbounded in every neighborhood (values grow without bound along the enumeration). The graph is closed in \mathbb{Q} \times \mathbb{R}: any convergent sequence (q_k, f(q_k)) \to (q, y) must have f(q_k) eventually constant (since otherwise the values diverge to infinity), implying the sequence is eventually constant at q, so y = f(q). These examples demonstrate that a closed graph does not suffice for continuity in general topological spaces. Additional conditions, such as compactness of the domain or completeness of both domain and codomain (as in the functional analytic version of the closed graph theorem), are necessary to ensure that mappings with closed graphs are continuous.^[12]

Topological Versions

General statement in point-set topology

In point-set topology, the closed graph theorem characterizes the relationship between continuity of a mapping and the closedness of its graph in the product space. Let X and Y be topological spaces with Y Hausdorff. If f: X \to Y is continuous, then the graph G_f = \{(x, f(x)) \mid x \in X\} is closed in the product topology on X \times Y. The converse holds under further assumptions on the spaces; for instance, if the graph G_f is closed and Y is compact, then f is continuous.^[13]^[14] To prove that continuity implies a closed graph, consider the complement \Omega = (X \times Y) \setminus G_f. For any (x_0, y_0) \in \Omega, it follows that y_0 \neq f(x_0). The Hausdorff property of Y yields disjoint open neighborhoods V of y_0 and W of f(x_0). Continuity of f ensures an open neighborhood U of x_0 such that f(U) \subseteq W. The product U \times V is then an open neighborhood of (x_0, y_0) contained in \Omega, showing that \Omega is open and hence G_f is closed.^[13] For the converse when Y is compact and Hausdorff, assume G_f is closed. To establish continuity at an arbitrary x \in X, let V be an open neighborhood of f(x) in Y. The set Y \setminus V is closed, so X \times (Y \setminus V) is closed in X \times Y. The intersection G_f \cap (X \times (Y \setminus V)) is thus closed, and its projection onto X under the natural projection map \pi_1: X \times Y \to X is closed because the projection is a closed map when the second factor is compact. This projection equals \{z \in X \mid f(z) \notin V\}, which excludes x. Hence, there exists an open neighborhood U of x disjoint from this projection, implying f(U) \subseteq V.^[14] When these conditions fail, such as when Y is not compact, the converse may not hold. A classic counterexample is the function f: \mathbb{R} \to \mathbb{R} defined by f(x) = 1/x for x \neq 0 and f(0) = 0. This function is discontinuous at $0 since \lim_{x \to 0} f(x) does not exist (while f(0) = 0), yet its graph is closed: if a sequence (x_n, f(x_n)) \to (a, b) in \mathbb{R}^2, then for a \neq 0, continuity of $1/x away from $0 ensures b = f(a); for a = 0, any convergent f(x_n) \to b would require x_n bounded away from $0 eventually (contradicting x_n \to 0), so no such limit point lies outside the graph.^[15] Recent generalizations extend the theorem beyond Hausdorff codomains, incorporating near-continuity and Baire-like properties on the spaces to infer continuity from closed graphs in non-Hausdorff settings.

Extension to set-valued functions

A set-valued function, or correspondence, f: X \to 2^Y, where X and Y are topological spaces and $2^Y denotes the power set of Y, assigns to each point in X a nonempty subset of Y. The graph of such a function is defined as G_f = \{ (x, y) \in X \times Y \mid y \in f(x) \} \subseteq X \times Y, generalizing the single-valued case where the graph consists of pairs (x, f(x)). In the context of set-valued functions, a key extension of the closed graph theorem establishes an equivalence between the closedness of the graph and upper hemicontinuity under suitable conditions. Specifically, if f is compact-valued (i.e., f(x) is compact for every x \in X) and Y is a Hausdorff space, then G_f is closed if and only if f is upper hemicontinuous. Upper hemicontinuity here means that for every sequence x_n \to x in X, the limit superior \limsup_{n \to \infty} f(x_n) \subseteq f(x), where \limsup_{n \to \infty} f(x_n) = \bigcap_{n=1}^\infty \overline{\bigcup_{k=n}^\infty f(x_k)} and the bar denotes closure in Y. This result holds in sequential terms for first-countable spaces but extends to the general topological setting via nets or filters.^[16]^[17] The proof proceeds in two directions. First, if G_f is closed, then upper hemicontinuity follows directly: suppose x_n \to x and y_n \in f(x_n) with y_n \to y; the sequence (x_n, y_n) converges to (x, y) in the product topology, so (x, y) \in G_f by closedness, implying y \in f(x) and thus the limsup inclusion. Conversely, if f is upper hemicontinuous and compact-valued, closedness of G_f is shown by contradiction: if (x_n, y_n) \to (x, y) with y_n \in f(x_n) but y \notin f(x), the compactness of the f(x_n) ensures a convergent subnet of y_n to some z \in \limsup f(x_n) \subseteq f(x), but Hausdorff separation in Y yields z = y, a contradiction unless y \in f(x). This relies crucially on compactness to extract convergent subsequences and control the limsup.^[17] Unlike the single-valued closed graph theorem, where closedness implies continuity without additional assumptions in metric spaces, the set-valued case requires compactness of the values to bridge closed graphs and upper hemicontinuity; without it, a closed graph may fail to ensure the limsup inclusion, as unbounded or non-compact sets can allow sequences escaping the limit set. For instance, non-compact values permit correspondences with closed graphs that are not upper hemicontinuous, highlighting the necessity of this condition for the equivalence. This extension finds application in economic theory, particularly in general equilibrium models where correspondences represent budget sets or demand mappings; upper hemicontinuous correspondences ensure stability of equilibria under perturbations, as in the analysis of fixed points for multivalued excess demand functions.

Functional Analytic Version

Statement for linear operators on Banach spaces

A Banach space is a complete normed vector space over the real or complex numbers, where completeness means that every Cauchy sequence converges in the space.^[18] A linear operator T: X \to Y between normed vector spaces X and Y is a function satisfying T(\alpha x + \beta z) = \alpha T(x) + \beta T(z) for all scalars \alpha, \beta and vectors x, z \in X. The graph of T is the subset G_T = \{ (x, T x) \mid x \in X \} of the product space X \times Y, equipped with the product norm \|(x, y)\| = \max(\|x\|, \|y\|), which is itself a Banach space when X and Y are.^[18] The closed graph theorem states that if X and Y are Banach spaces and T: X \to Y is a linear operator whose graph G_T is closed in X \times Y, then T is continuous, or equivalently, bounded.^[18] This means there exists a constant M > 0 such that \|T x\| \leq M \|x\| for all x \in X.^[19] The theorem is a special case of more general topological versions but specifies the analytic structure of complete normed spaces.^[20] The completeness of the Banach spaces is crucial, as it leverages properties like the Baire category theorem to link the topological closedness of the graph to uniform boundedness of the operator, a connection that fails in incomplete normed spaces.^[18] Without completeness, there exist linear operators with closed graphs that are unbounded, highlighting the necessity of the Banach space setting.^[21] An example of an unbounded linear operator whose graph is not closed arises in the context of differentiation with a dense domain. Consider the space X = C[0,1] of continuous functions on [0,1] with the supremum norm, which is a Banach space. Let P be the dense subspace of polynomials in X, and define the differentiation operator D: P \to X by D(p) = p'. This operator is linear but unbounded, as high-degree polynomials can have derivatives with norms much larger than the original. The graph G_D = \{ (p, p') \mid p \in P \} \subset X \times X is not closed; for instance, Bernstein polynomials approximating a non-differentiable continuous function converge in X, but their derivatives do not converge to a continuous limit in X, so limit points lie outside G_D.^[22] The theorem was proved by Stefan Banach in 1932 as part of the foundational development of operator theory in normed linear spaces.^[23]

Proof sketch using open mapping theorem

The proof assumes that X and Y are Banach spaces and T: X \to Y is a linear operator whose graph G(T) = \{(x, Tx) \mid x \in X\} is closed in the product space X \times Y. Since X \times Y is Banach under the product norm and G(T) is a closed subspace, G(T) is itself a Banach space.^[1] Consider the projection \pi_X: G(T) \to X defined by \pi_X(x, Tx) = x. This map is linear and continuous, as it is the restriction of the continuous projection from X \times Y to X, and it is bijective onto X. The open mapping theorem states that any surjective continuous linear operator between Banach spaces is open, implying that the inverse \pi_X^{-1}: X \to G(T) is continuous.^[24] The operator T can then be factored as T = \pi_Y \circ \pi_X^{-1}, where \pi_Y: G(T) \to Y is the continuous projection \pi_Y(x, Tx) = Tx. As a composition of continuous linear operators, T is continuous.^[18] An alternative proof approach invokes the Baire category theorem directly to establish continuity without relying on the open mapping theorem.^[1] This result hinges on the completeness of the spaces; the theorem fails in incomplete normed spaces, where discontinuous linear operators with closed graphs exist.^[21]

Connections and Generalizations

Relation to the open mapping theorem

The closed graph theorem and the open mapping theorem are closely related in the context of Banach spaces, with each implying the other. The closed graph theorem follows as a corollary of the open mapping theorem: for a linear operator T: X \to Y between Banach spaces with closed graph, the projection from the graph to X is a continuous linear bijection, and by the open mapping theorem, its inverse is continuous, implying T is bounded.^[14] Conversely, the open mapping theorem can be derived from the closed graph theorem by considering the quotient space construction. Let T: X \to Y be a bounded surjective linear operator between Banach spaces X and Y. Define the kernel N(T) = \{x \in X : Tx = 0\}, which is closed, and the quotient space X/N(T) with the quotient norm. The induced operator G: Y \to X/N(T) given by G(y) = , where x \in X satisfies Tx = y, has a closed graph: if y_n \to y and G(y_n) \to in X/N(T), then there exist z_n \in N(T) such that Tx_n = y_n and [x_n - x] \to 0, so Tx = y by continuity of T. By the closed graph theorem, G is bounded, meaning there exists \delta > 0 such that B_Y(0, \delta) \subset G(B_{X/N(T)}(0, 1)), which implies B_Y(0, \delta) \subset T(B_X(0, 1)) and thus T is open. The closed graph theorem is also equivalent to the uniform boundedness principle in the context of Banach spaces.^[25] Both theorems emerged in the 1930s, with the open mapping theorem stated by Stefan Banach in his 1932 monograph Théorie des opérations linéaires, and the closed graph theorem developed concurrently, including early versions for Hilbert spaces by John von Neumann.^[26] Similar equivalences hold in more general settings, such as Fréchet spaces or locally convex topological vector spaces, where bounded surjective operators with closed graphs satisfy open mapping properties under completeness assumptions. Recent work extends these relations to metric spaces, providing characterizations of spaces where closed graph and open mapping principles coincide.^[27]

Applications in operator theory

The closed graph theorem plays a pivotal role in operator theory by establishing the boundedness of certain linear operators on Hilbert or Banach spaces, particularly when they are defined everywhere. A key application is the Hellinger–Toeplitz theorem, which asserts that any symmetric linear operator defined on the entire Hilbert space \mathcal{H} is bounded. The proof relies on showing that the graph of such an operator is closed—due to the symmetry condition \langle Tx, y \rangle = \langle x, Ty \rangle for all x, y \in \mathcal{H}—and then applying the closed graph theorem to conclude boundedness.^[22] This result underscores that unbounded self-adjoint operators cannot be defined on the full space, motivating the construction of self-adjoint extensions for densely defined symmetric operators with proper domains, as in the theory of differential operators where the minimal operator is extended to a self-adjoint one via deficiency indices.^[22] In spectral theory, the closed graph theorem ensures the boundedness of resolvent operators, which is essential for analyzing the spectrum of closed operators. For a closed densely defined operator T on a Banach space X, if \lambda belongs to the resolvent set \rho(T), then the resolvent R(\lambda, T) = (T - \lambda I)^{-1} is defined on all of X and has a closed graph, implying it is bounded by the closed graph theorem.^[28] This boundedness facilitates the holomorphic dependence of the resolvent on \lambda and the continuity of the spectrum with respect to perturbations, as seen in the Riesz–Dunford calculus for unbounded operators.^[28] A concrete example arises in harmonic analysis, where the closed graph theorem confirms the boundedness of the Fourier transform on L^p(\mathbb{R}^n) spaces for $1 < p < \infty. The Fourier transform is initially defined and continuous on the dense subspace of Schwartz functions, and its extension to all of L^p has a closed graph—since convergence in L^p implies convergence of transforms in appropriate senses—yielding the operator norm bound \|\hat{f}\|_{L^{p'}} \leq C \|f\|_{L^p} (with $1/p + 1/p' = 1) via the closed graph theorem.^[29] Generalizations of the closed graph theorem extend to unbounded operators through domain closure: the closure \overline{T} of a closable operator T is the unique closed extension whose graph is the closure of the graph of T, preserving linearity and enabling the study of maximal extensions in Hilbert spaces. In control theory, the theorem adapts to set-valued linear operators modeled as closed convex processes, where a closed graph implies continuity in the sense of bounded multimaps, facilitating analysis of controllability and reachability sets for systems like differential inclusions.^[30]

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