Closed graph theorem (functional analysis)

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[]

The graph of a function ${\displaystyle f:X\to Y}$ is the set

$${\displaystyle \operatorname {Gr} f:=\{(x,f(x)):x\in X\}=\{(x,y)\in X\times Y:y=f(x)\}.}$$

As usual, if ${\displaystyle X}$ and ${\displaystyle Y}$ are topological spaces then ${\displaystyle X\times Y}$ is assumed to be endowed with the product topology.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are topological spaces, ${\displaystyle D\subseteq X,}$ and ${\displaystyle f:D\to Y}$ is a function, then ${\displaystyle f}$ has a closed graph (resp. sequentially closed graph) in ${\displaystyle X\times Y}$ if the graph of ${\displaystyle f,}$ ${\displaystyle \operatorname {Gr} f,}$ is a closed (resp. sequentially closed) subset of ${\displaystyle X\times Y.}$ If ${\displaystyle D=X}$ or if ${\displaystyle X}$ is clear from context then "in ${\displaystyle X\times Y}$" may be omitted from writing.

Linear operators[]

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

A map ${\displaystyle f:D\subseteq X\to Y}$ is closed (resp. sequentially closed) or has a closed graph (resp. has a sequentially closed graph) if the graph of ${\displaystyle f}$ is closed (resp. sequentially closed) in ${\displaystyle X\times Y}$ (rather than in ${\displaystyle D\times Y}$).

A map ${\displaystyle f:D\subseteq X\to Y}$ is a linear or a linear operator if ${\displaystyle X}$ and ${\displaystyle Y}$ are vector spaces, ${\displaystyle D\subseteq X}$ is a vector subspace of ${\displaystyle X,}$ ${\displaystyle f:D\to Y}$ is a linear map.

Closed linear operators[]

Assume that ${\displaystyle X}$ and ${\displaystyle Y}$ are topological vector spaces (TVSs).

A linear operator ${\displaystyle f:D\subseteq X\to Y}$ is called closed or a closed linear operator if its graph is closed in ${\displaystyle X\times Y.}$

Closable maps and closures

A linear operator ${\displaystyle f:D\subseteq X\to Y}$ is closable in ${\displaystyle X\times Y}$ if there exists a vector subspace ${\displaystyle E\subseteq X}$ containing ${\displaystyle D}$ and a function (resp. multifunction) ${\displaystyle F:E\to Y}$ whose graph is equal to the closure of the set ${\displaystyle \operatorname {Gr} f}$ in ${\displaystyle X\times Y.}$ Such an ${\displaystyle F}$ is called a closure of ${\displaystyle f}$ in ${\displaystyle X\times Y}$, is denoted by ${\displaystyle {\overline {f}},}$ and necessarily extends ${\displaystyle f.}$

If ${\displaystyle f:D\subseteq X\to Y}$ is a closable linear operator then a core or an essential domain of ${\displaystyle f}$ is a subset ${\displaystyle C\subseteq D}$ such that the closure in ${\displaystyle X\times Y}$ of the graph of the restriction ${\displaystyle f{\big \vert }_{C}:C\to Y}$ of ${\displaystyle f}$ to ${\displaystyle C}$ is equal to the closure of the graph of ${\displaystyle f}$ in ${\displaystyle X\times Y}$ (i.e. the closure of ${\displaystyle \operatorname {Gr} f}$ in ${\displaystyle X\times Y}$ is equal to the closure of ${\displaystyle \operatorname {Gr} f{\big \vert }_{C}}$ in ${\displaystyle X\times Y}$).

Closed maps vs. closed linear operators

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

A map ${\displaystyle f:X\to Y}$ between topological spaces is called a closed map if the image of a closed subset of ${\displaystyle X}$ is a closed subset of ${\displaystyle Y.}$

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.

Characterizations of closed graphs (general topology)[]

Throughout, let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological spaces and ${\displaystyle X\times Y}$ is endowed with the product topology.

Function with a closed graph

If ${\displaystyle f:X\to Y}$ is a function then it is said to have a closed graph if it satisfies any of the following are equivalent conditions:

1. (Definition): The graph ${\displaystyle \operatorname {Gr} f}$ of ${\displaystyle f}$ is a closed subset of ${\displaystyle X\times Y.}$
2. For every ${\displaystyle x\in X}$ and net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in ${\displaystyle X}$ such that ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle X,}$ if ${\displaystyle y\in Y}$ is such that the net ${\displaystyle f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i\in I}}$ in ${\displaystyle Y}$ then ${\displaystyle y=f(x).}$^[2]
   • Compare this to the definition of continuity in terms of nets, which recall is the following: for every ${\displaystyle x\in X}$ and net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in ${\displaystyle X}$ such that ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle X,}$ ${\displaystyle f\left(x_{\bullet }\right)\to f(x)}$ in ${\displaystyle Y.}$
   • Thus to show that the function ${\displaystyle f}$ has a closed graph, it may be assumed that ${\displaystyle f\left(x_{\bullet }\right)}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y}$ (and then show that ${\displaystyle y=f(x)}$) while to show that ${\displaystyle f}$ is continuous, it may not be assumed that ${\displaystyle f\left(x_{\bullet }\right)}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y}$ and instead, it must be proven that this is true (and moreover, it must more specifically be proven that ${\displaystyle f\left(x_{\bullet }\right)}$ converges to ${\displaystyle f(x)}$ in ${\displaystyle Y}$).

and if ${\displaystyle Y}$ is a Hausdorff compact space then we may add to this list:

3. ${\displaystyle f}$ is continuous.^[3]

and if both ${\displaystyle X}$ and ${\displaystyle Y}$ are first-countable spaces then we may add to this list:

4. ${\displaystyle f}$ has a sequentially closed graph in ${\displaystyle X\times Y.}$

Function with a sequentially closed graph

If ${\displaystyle f:X\to Y}$ is a function then the following are equivalent:

1. ${\displaystyle f}$ has a sequentially closed graph in ${\displaystyle X\times Y.}$
2. Definition: the graph of ${\displaystyle f}$ is a sequentially closed subset of ${\displaystyle X\times Y.}$
3. For every ${\displaystyle x\in X}$ and sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ such that ${\displaystyle x_{\bullet }\to x}$ in ${\displaystyle X,}$ if ${\displaystyle y\in Y}$ is such that the net ${\displaystyle f\left(x_{\bullet }\right):=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }\to y}$ in ${\displaystyle Y}$ then ${\displaystyle y=f(x).}$^[2]

Basic properties of maps with closed graphs[]

Suppose ${\displaystyle f:D(f)\subseteq X\to Y}$ is a linear operator between Banach spaces.

• If ${\displaystyle A}$ is closed then ${\displaystyle A-s\operatorname {Id} _{D(f)}}$ is closed where ${\displaystyle s}$ is a scalar and ${\displaystyle \operatorname {Id} _{D(f)}}$ is the identity function.
• If ${\displaystyle f}$ is closed, then its kernel (or nullspace) is a closed vector subspace of ${\displaystyle X.}$
• If ${\displaystyle f}$ is closed and injective then its inverse ${\displaystyle f^{-1}}$ is also closed.
• A linear operator ${\displaystyle f}$ admits a closure if and only if for every ${\displaystyle x\in X}$ and every pair of sequences ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ and ${\displaystyle z_{\bullet }=\left(z_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle D(f)}$ both converging to ${\displaystyle x}$ in ${\displaystyle X,}$ such that both ${\displaystyle f\left(x_{\bullet }\right)=\left(f\left(x_{i}\right)\right)_{i=1}^{\infty }}$ and ${\displaystyle f\left(z_{\bullet }\right)=\left(f\left(z_{i}\right)\right)_{i=1}^{\infty }}$ converge in ${\displaystyle Y,}$ one has ${\displaystyle \lim _{i\to \infty }f\left(x_{i}\right)=\lim _{i\to \infty }f\left(z_{i}\right).}$

Examples and counterexamples[]

Continuous but not closed maps[]

• Let ${\displaystyle X}$ denote the real numbers ${\displaystyle \mathbb {R} }$ with the usual Euclidean topology and let ${\displaystyle Y}$ denote ${\displaystyle \mathbb {R} }$ with the indiscrete topology (where ${\displaystyle Y}$ is not Hausdorff and that every function valued in ${\displaystyle Y}$ is continuous). Let ${\displaystyle f:X\to Y}$ be defined by ${\displaystyle f(0)=1}$ and ${\displaystyle f(x)=0}$ for all ${\displaystyle x\neq 0.}$ Then ${\displaystyle f:X\to Y}$ is continuous but its graph is not closed in ${\displaystyle X\times Y.}$^[2]
• If ${\displaystyle X}$ is any space then the identity map ${\displaystyle \operatorname {Id} :X\to X}$ is continuous but its graph, which is the diagonal ${\displaystyle \operatorname {Gr} \operatorname {Id} =\{(x,x):x\in X\},}$ is closed in ${\displaystyle X\times X}$ if and only if ${\displaystyle X}$ is Hausdorff.^[4] In particular, if ${\displaystyle X}$ is not Hausdorff then ${\displaystyle \operatorname {Id} :X\to X}$ is continuous but not closed.
• If ${\displaystyle f:X\to Y}$ is a continuous map whose graph is not closed then ${\displaystyle Y}$ is not a Hausdorff space.

Closed but not continuous maps[]

• If ${\displaystyle (X,\tau )}$ is a Hausdorff TVS and ${\displaystyle \nu }$ is a vector topology on ${\displaystyle X}$ that is strictly finer than ${\displaystyle \tau ,}$ then the identity map ${\displaystyle \operatorname {Id} :(X,\tau )\to (X,\nu )}$ a closed discontinuous linear operator.^[5]
• Consider the derivative operator ${\displaystyle A={\frac {d}{dx}}}$ where ${\displaystyle X=Y=C([a,b]).}$is the Banach space of all continuous functions on an interval ${\displaystyle [a,b].}$ If one takes its domain ${\displaystyle D(f)}$ to be ${\displaystyle C^{1}([a,b]),}$ then ${\displaystyle f}$ is a closed operator, which is not bounded.^[6] On the other hand, if D(f) = ${\displaystyle C^{\infty }([a,b]).}$, then ${\displaystyle f}$ will no longer be closed, but it will be closable, with the closure being its extension defined on ${\displaystyle C^{1}([a,b]).}$
• Let ${\displaystyle X}$ and ${\displaystyle Y}$ both denote the real numbers ${\displaystyle \mathbb {R} }$ with the usual Euclidean topology. Let ${\displaystyle f:X\to Y}$ be defined by ${\displaystyle f(0)=0}$ and ${\displaystyle f(x)={\frac {1}{x}}}$ for all ${\displaystyle x\neq 0.}$ Then ${\displaystyle f:X\to Y}$ has a closed graph (and a sequentially closed graph) in ${\displaystyle X\times Y=\mathbb {R} ^{2}}$ but it is not continuous (since it has a discontinuity at ${\displaystyle x=0}$).^[2]
• Let ${\displaystyle X}$ denote the real numbers ${\displaystyle \mathbb {R} }$ with the usual Euclidean topology, let ${\displaystyle Y}$ denote ${\displaystyle \mathbb {R} }$ with the discrete topology, and let ${\displaystyle \operatorname {Id} :X\to Y}$ be the identity map (i.e. ${\displaystyle \operatorname {Id} (x):=x}$ for every ${\displaystyle x\in X}$). Then ${\displaystyle \operatorname {Id} :X\to Y}$ is a linear map whose graph is closed in ${\displaystyle X\times Y}$ but it is clearly not continuous (since singleton sets are open in ${\displaystyle Y}$ but not in ${\displaystyle X}$).^[2]

Closed graph theorems[]

Between Banach spaces[]

In functional analysis, the closed graph theorem states the following: If ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces, and ${\displaystyle T:X\to Y}$ is a linear operator, then ${\displaystyle T}$ is continuous if and only if its graph is closed in ${\displaystyle X\times Y}$ (with the product topology).

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

The operator is required to be everywhere-defined, that is, the domain ${\displaystyle D(T)}$ of ${\displaystyle T}$ is ${\displaystyle X.}$ 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 ${\displaystyle C([0,1]),}$ whose domain is a strict subset of ${\displaystyle C([0,1]).}$

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 ${\displaystyle X}$ and ${\displaystyle Y}$ 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.

Complete metrizable codomain[]

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

Between F-spaces[]

There are versions that does not require ${\displaystyle Y}$ to be locally convex.

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

Complete pseudometrizable codomain[]

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

Codomain not complete or (pseudo) metrizable[]

An even more general version of the closed graph theorem is

Borel graph theorem[]

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:

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

A topological space ${\displaystyle X}$ is called a K_σδ if it is the countable intersection of countable unions of compact sets.

A Hausdorff topological space ${\displaystyle Y}$ is called K-analytic if it is the continuous image of a K_σδ space (that is, if there is a K_σδ space ${\displaystyle X}$ and a continuous map of ${\displaystyle X}$ onto ${\displaystyle Y}$).

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:

Related results[]

If ${\displaystyle F:X\to Y}$ is closed linear operator from a Hausdorff locally convex TVS ${\displaystyle X}$ into a Hausdorff finite-dimensional TVS ${\displaystyle Y}$ then F is continuous.^[15]

See also[]

• 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

References[]

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