Closed graph theorem

Last updated
Dirac distribution CDF.svg
The graph of the cubic function f(x) = x3 − 9x on the interval [−4,4] is closed because the function is continuous. The graph of the Heaviside function on [−2,2] is not closed, because the function is not continuous.

In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs. In particular, they give conditions when functions with closed graphs are necessarily continuous. In mathematics, there are several results known as the "closed graph theorem".


Graphs and maps with closed graphs

If f : XY is a map between topological spaces then the graph of f is the set Gr f := { (x, f(x)) : xX} or equivalently,

Gr f := { (x, y) ∈ X × Y : y = f(x)}

We say that the graph of f is closed if Gr f is a closed subset of X×Y (with the product topology).

Any continuous function into a Hausdorff space has a closed graph.

Any linear map, L : XY, between two topological vector spaces whose topologies are (Cauchy) complete with respect to translation invariant metrics, and if in addition (1a) L is sequentially continuous in the sense of the product topology, then the map L is continuous and its graph, Gr L, is necessarily closed. Conversely, if L is such a linear map with, in place of (1a), the graph of L is (1b) known to be closed in the Cartesian product space X×Y, then L is continuous and therefore necessarily sequentially continuous. [1]

Examples of continuous maps that are not closed

Closed graph theorem in point-set topology

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

Closed graph theorem [4]   If f : XY is a map from a topological space X into a compact Hausdorff space Y, then the graph of f is closed if and only if f : XY is continuous.

For set-valued functions

Closed graph theorem for set-valued functions [5]   For a Hausdorff compact range space Y, a set-valued function F : X → 2Y has a closed graph if and only if it is upper hemicontinuous and F(x) is a closed set for all xX.

In functional analysis

Definition: If T : XY is a linear operator between topological vector spaces (TVSs) then we say that T is a closed operator if the graph of T is closed in X × Y when X × Y 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.

Theorem [6] [7]   A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.

See also

Related Research Articles

In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.

Functional analysis Branch of mathematical analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed vector space to make the study of the dual space "interesting". Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry.

Normed vector space Vector space on which a distance is defined

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function defined on the vector space that is commonly denoted and has the following properties:

  1. It is nonnegative, that is for every vector x, one has
  2. It is positive on nonzero vectors, that is,
  3. For every vector x, and every scalar one has
  4. The triangle inequality holds; that is, for every vectors x and y, one has

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → ℝ so that

  1. Scalar multiplication in V is continuous with respect to d and the standard metric on ℝ or ℂ.
  2. Addition in V is continuous with respect to d.
  3. The metric is translation-invariant; i.e., d(x + a, y + a) = d(x, y) for all x, y and a in V.
  4. The metric space (V, d) is complete.

In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from X into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. Note that in 1951, R. C. James discovered a non-reflexive Banach space that is isometrically isomorphic to its bidual.

In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional analysis and related areas of mathematics, a barrelled space is a topological vector spaces (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed. Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.

In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem, is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.

In mathematics, particularly in functional analysis, a webbed space is a topological vector space designed with the goal of allowing the results of the open mapping theorem and the closed graph theorem to hold for a wider class of linear maps whose codomains are webbed spaces. A space is called webbed if there exists a collection of sets, called a web that satisfies certain properties. Webs were first investigated by de Wilde.

In mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : XY between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.

In functional analysis, a topological vector space (TVS) is called ultrabornological if every bounded linear operator from into another TVS is necessarily continuous. A general version of the closed graph theorem holds for ultrabornological spaces. Ultrabornological spaces were introduced by Alexander Grothendieck.

This is a glossary for the terminology in a mathematical field of functional analysis.

F. Riesz's theorem is an important theorem in functional analysis that states that a Hausdorff topological vector space (TVS) is finite-dimensional if and only if it is locally compact. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

In mathematics, specifically in functional analysis and Hilbert space theory, vector-valued Hahn–Banach theorems are generalizations of the Hahn–Banach theorems from linear functionals to linear operators valued in topological vector spaces (TVSs).

In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."

In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map.


  1. Rudin 1991, p. 51-52.
  2. Rudin 1991, p. 50.
  3. Narici & Beckenstein 2011, pp. 459-483.
  4. Munkres 2000, pp. 163–172.
  5. Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17". Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer.
  6. Schaefer & Wolff 1999, p. 78.
  7. Trèves (1995) , p. 173