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."

- Definitions
- Graphs and closed graphs
- Linear operators
- Closed linear operators
- Characterizations of closed graphs (general topology)
- Basic properties of maps with closed graphs
- Examples and counterexamples
- Continuous but not closed maps
- Closed but not continuous maps
- Closed graph theorems
- Between Banach spaces
- Complete metrizable codomain
- Complete pseudometrizable codomain
- Codomain not complete or (pseudo) metrizable
- Borel graph theorem
- Related results
- See also
- References
- Bibliography

The ** graph ** of a function is the set

As usual, if and are topological spaces then is assumed to be endowed with the product topology.

If and are topological spaces, and is a function, then has a **closed graph** (resp. **sequentially closed graph**) **in ** if the graph of is a closed (resp. sequentially closed) subset of If or if is clear from context then "in " may be omitted from writing.

A ** partial map **,^{ [1] } denoted by *f* : *X* ↣ *Y*, if a map from a subset of denoted by dom *f*, into If is written then it is meant that *f* : *X* ↣ *Y* is a partial map and dom *f* = *D*.

A map is **closed** (resp. **sequentially closed**) or **has a closed graph** (resp. **has a sequentially closed graph**) if the graph of is closed (resp. sequentially closed) in (rather than in ).

A map is a **linear** or a **linear operator** if and are vector spaces, is a vector subspace of is a linear map.

Assume that and are topological vector spaces (TVSs).

A linear operator is called **closed** or a **closed linear operator** if its graph is closed in

- Closable maps and closures

A linear operator is **closable in ** if there exists a *vector subspace* containing and a function (resp. multifunction) whose graph is equal to the closure of the set in Such an is called a **closure of in **, is denoted by and necessarily extends

If is a closable linear operator then a **core** or an **essential domain** of is a subset such that the closure in of the graph of the restriction of to is equal to the closure of the graph of in (i.e. the closure of in is equal to the closure of in ).

- Closed maps vs. closed linear operators

When reading literature in functional analysis, if is a linear map between topological vector spaces (TVSs) then " is closed" will almost always mean that its graph is closed. However, " is closed" may, especially in literature about point-set topology, instead mean the following:

A map between topological spaces is called a ** closed map ** if the image of a closed subset of is a closed subset of

These two definitions of "closed map" are not equivalent. If it is unclear, then it is recommended that a reader check how "closed map" is defined by the literature they are reading.

Throughout, let and be topological spaces and is endowed with the product topology.

- Function with a closed graph

If is a function then it is said to have a ** closed graph** if it satisfies any of the following are equivalent conditions:

- (Definition): The graph of is a closed subset of
- For every and net in such that in if is such that the net in then
^{ [2] }- Compare this to the definition of continuity in terms of nets, which recall is the following: for every and net in such that in in
- Thus to show that the function has a closed graph, it
*may*be assumed that converges in to some (and then show that ) while to show that is continuous, it may*not*be assumed that converges in to some and instead, it must be proven that this is true (and moreover, it must more specifically be proven that converges to in ).

and if is a Hausdorff compact space then we may add to this list:

- is continuous.
^{ [3] }

and if both and are first-countable spaces then we may add to this list:

- has a sequentially closed graph in

- Function with a sequentially closed graph

If is a function then the following are equivalent:

- has a sequentially closed graph in
- Definition: the graph of is a sequentially closed subset of
- For every and sequence in such that in if is such that the net in then
^{ [2] }

Suppose is a linear operator between Banach spaces.

- If is closed then is closed where is a scalar and is the identity function.
- If is closed, then its kernel (or nullspace) is a closed vector subspace of
- If is closed and injective then its inverse is also closed.
- A linear operator admits a closure if and only if for every and every pair of sequences and in both converging to in such that both and converge in one has

- Let denote the real numbers with the usual Euclidean topology and let denote with the indiscrete topology (where is
*not*Hausdorff and that every function valued in is continuous). Let be defined by and for all Then is continuous but its graph is*not*closed in^{ [2] } - If is any space then the identity map is continuous but its graph, which is the diagonal is closed in if and only if is Hausdorff.
^{ [4] }In particular, if is not Hausdorff then is continuous but*not*closed. - If is a continuous map whose graph is not closed then is
*not*a Hausdorff space.

- If is a Hausdorff TVS and is a vector topology on that is strictly finer than then the identity map a closed discontinuous linear operator.
^{ [5] } - Consider the derivative operator where is the Banach space of all continuous functions on an interval If one takes its domain to be then is a closed operator, which is not bounded.
^{ [6] }On the other hand, if*D*(*f*) = , then will no longer be closed, but it will be closable, with the closure being its extension defined on - Let and both denote the real numbers with the usual Euclidean topology. Let be defined by and for all Then has a closed graph (and a sequentially closed graph) in but it is
*not*continuous (since it has a discontinuity at ).^{ [2] } - Let denote the real numbers with the usual Euclidean topology, let denote with the discrete topology, and let be the identity map (i.e. for every ). Then is a linear map whose graph is closed in but it is clearly
*not*continuous (since singleton sets are open in but not in ).^{ [2] }

In functional analysis, the closed graph theorem states the following: If and are Banach spaces, and is a linear operator, then is continuous if and only if its graph is closed in (with the product topology).

The closed graph theorem can be reformulated may be rewritten into a form that is more easily usable:

**Closed Graph Theorem for Banach spaces** — If is a linear operator between Banach spaces, then the following are equivalent:

- is continuous.
- is a closed operator (that is, the graph of is closed).
- If in then in
- If in then in
- If in and if converges in to some then
- If in and if converges in to some then

The operator is required to be **everywhere-defined**, that is, the domain of is This condition is necessary, as there exist closed linear operators that are unbounded (not continuous); a prototypical example is provided by the derivative operator on whose domain is a strict subset of

The usual proof of the closed graph theorem employs the open mapping theorem. In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem are all equivalent. This equivalence also serves to demonstrate the importance of and being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.

The closed graph theorem can be generalized from Banach spaces to more abstract topological vector spaces in the following ways.

**Theorem** — A linear operator from a barrelled space to a Fréchet space is continuous if and only if its graph is closed.

There are versions that does not require to be locally convex.

**Theorem** — A linear map between two F-spaces is continuous if and only if its graph is closed.^{ [7] }^{ [8] }

This theorem is restated and extend it with some conditions that can be used to determine if a graph is closed:

**Theorem** — If is a linear map between two F-spaces, then the following are equivalent:

- is continuous.
- has a closed graph.
- If in and if converges in to some then
^{ [9] } - If in and if converges in to some then

Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.

**Closed Graph Theorem ^{ [10] }** — Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous.

**Closed Graph Theorem** — A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous.^{ [10] }

**Theorem ^{ [11] }** — Suppose that is a linear map whose graph is closed. If is an inductive limit of Baire TVSs and is a webbed space then is continuous.

**Closed Graph Theorem ^{ [10] }** — A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous.

An even more general version of the closed graph theorem is

**Theorem ^{ [12] }** — Suppose that and are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

- If is any closed subspace of and is any continuous map of onto then is an open mapping.

Under this condition, if is a linear map whose graph is closed then is continuous.

The Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis.^{ [13] } Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space. The weak dual of a separable Fréchet space and the strong dual of a separable Fréchet-Montel space are Souslin spaces. Also, the space of distributions and all Lp-spaces over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem states:

**Borel Graph Theorem** — Let be linear map between two locally convex Hausdorff spaces and If is the inductive limit of an arbitrary family of Banach spaces, if is a Souslin space, and if the graph of u is a Borel set in then u is continuous.^{ [13] }

An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces.

A topological space is called a K_{σδ} if it is the countable intersection of countable unions of compact sets.

A Hausdorff topological space is called **K-analytic** if it is the continuous image of a K_{σδ} space (that is, if there is a K_{σδ} space and a continuous map of onto ).

Every compact set is K-analytic so that there are non-separable K-analytic spaces. Also, every Polish, Souslin, and reflexive Fréchet space is K-analytic as is the weak dual of a Frechet space. The generalized Borel graph theorem states:

**Generalized Borel Graph Theorem ^{ [14] }** — Let be a linear map between two locally convex Hausdorff spaces and If is the inductive limit of an arbitrary family of Banach spaces, if is a K-analytic space, and if the graph of u is closed in then u is continuous.

If is closed linear operator from a Hausdorff locally convex TVS into a Hausdorff finite-dimensional TVS then F is continuous.^{ [15] }

- Almost open linear map
- Banach space – Normed vector space that is complete
- Barrelled space – A topological vector space with near minimum requirements for the Banach–Steinhaus theorem to hold.
- Closed graph – Graph of a function that is also a closed subset of the product space
- Closed linear operator
- Continuous linear map
- Densely defined operator – Function that is defined almost everywhere (mathematics)
- Discontinuous linear map
- Kakutani fixed-point theorem – On when a function f: S→Pow(S) on a compact nonempty convex subset S⊂ℝⁿ has a fixed point
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Open mapping theorem (functional analysis) – Theorem giving conditions for a continuous linear map to be an open map
- Topological vector space – Vector space with a notion of nearness
- Ursescu theorem – Theorem that simultaneously generalizes the closed graph, open mapping, and Banach–Steinhaus theorems
- Webbed space – Topological vector spaces for which the open mapping and closed graphs theorems hold

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.

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 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 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 is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is *not* reflexive but is nevertheless 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 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".

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 branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

In functional analysis and related areas of mathematics, a **sequence space** is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field *K* of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in *K*, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In functional analysis and related areas of mathematics, a **barrelled space** is a topological vector space (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 mathematics, particularly in functional analysis, a **bornological space** is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a topological space possesses the minimum amount of structure needed to address questions of continuity. Bornological spaces are distinguished by that property that a linear map from a bornological space into any locally convex spaces is continuous if and only if it is a bounded linear operator.

In mathematics, **nuclear spaces** are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, another generalization of finite dimensional Euclidean spaces. They were introduced by Alexander Grothendieck.

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 functional analysis and related areas of mathematics, a **complete topological vector space** is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by *Cauchy nets* or *Cauchy filters*, which are generalizations of *Cauchy sequences*, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for *all* TVSs, including those that are not metrizable or Hausdorff.

In the branch of mathematics called functional analysis, a **complemented subspace** of a normed space or more generally, of a topological vector space is a vector subspace for which there exists some other vector subspace of called its (**topological**) **complement** in that allows to be treated as if it were the direct sum or product both algebraically and topologically; a situation that is summarized by saying that is the **topological direct sum** of these subspace or that is their direct sum in the category of topological vector spaces. Specifically, this means that is the algebraic direct sum of and also that the addition map that sends is a homeomorphism.

The theorem on the **surjection of Fréchet spaces** is an important theorem, due to Stefan Banach, that characterizes when a continuous linear operator between Fréchet spaces is surjective.

In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces. One method is used if the disk is bounded: in this case, the **auxiliary normed space** is with norm The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic.

In mathematics, **nuclear operators** are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

In functional analysis, a **topological homomorphism** or simply **homomorphism** is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional analysis and the famous open mapping theorem gives a sufficient condition for a continuous linear map between Fréchet spaces to be a topological homomorphism.

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

In functional analysis and related areas of mathematics, a **metrizable** topological vector space (TVS) is a TVS whose topology is induced by a metric. An **LM-space** is an inductive limit of a sequence of locally convex metrizable TVS.

- ↑ Dolecki & Mynard 2016, pp. 4-5.
- 1 2 3 4 5 Narici & Beckenstein 2011, pp. 459-483.
- ↑ Munkres 2000, p. 171.
- ↑ Rudin 1991, p. 50.
- ↑ Narici & Beckenstein 2011, p. 480.
- ↑ Kreyszig, Erwin (1978).
*Introductory Functional Analysis With Applications*. USA: John Wiley & Sons. Inc. p. 294. ISBN 0-471-50731-8. - ↑ Schaefer & Wolff 1999, p. 78.
- ↑ Trèves (1995) , p. 173
- ↑ Rudin 1991, pp. 50-52.
- 1 2 3 Narici & Beckenstein 2011, pp. 474-476.
- ↑ Narici & Beckenstein 2011, p. 479-483.
- ↑ Trèves 2006, p. 169.
- 1 2 Trèves 2006, p. 549.
- ↑ Trèves 2006, pp. 557-558.
- ↑ Narici & Beckenstein 2011, p. 476.

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