Closed graph theorem (functional analysis)

Let $T\; :\; X\; \backslash to\; Y$ be a linear operator between Banach spaces (or more generally Fréchet spaces). Then the continuity of $T$ means that $Tx\_i\; \backslash to\; Tx$ for each convergent sequence $x\_i\; \backslash to\; x$. On the other hand, the closedness of the graph of $T$ means that for each convergent sequence $x\_i\; \backslash to\; x$ such that $Tx\_i\; \backslash to\; y$, we have $y\; =\; Tx$. Hence, the closed graph theorem says that in order to check the continuity of $T$, one can show $T\; x\_i\; \backslash to\; Tx$ under the additional assumption that $Tx\_i$ is convergent.

In fact, for the graph of T to be closed, it is enough that if $x\_i\; \backslash to\; 0,\; \backslash ,\; Tx\_i\; \backslash to\; y$, then $y\; =\; 0$. Indeed, assuming that condition holds, if $(x\_i,\; Tx\_i)\; \backslash to\; (x,\; y)$, then $x\_i\; -\; x\; \backslash to\; 0$ and $T(x\_i\; -\; x)\; \backslash to\; y\; -\; Tx$. Thus, $y\; =\; Tx$; i.e., $(x,\; y)$ is in the graph of T.

Note, to check the closedness of a graph, it’s not even necessary to use the norm topology: if the graph of T is closed in some topology coarser than the norm topology, then it is closed in the norm topology.Theorem 4 of Tao. NB: The Hausdorffness there is put to ensure the graph of a continuous map is closed. In practice, this works like this: T is some operator on some function space. One shows T is continuous with respect to the distribution topology; thus, the graph is closed in that topology, which implies closedness in the norm topology and then T is a bounded by the closed graph theorem (when the theorem applies). See {{section link||Example}} for an explicit example.

{{Math theorem|math_statement= {{harvnb|Vogt|2000|loc=Theorem 1.8.}}

If $T\; :\; X\; \backslash to\; Y$ is a linear operator between Banach spaces (or more generally Fréchet spaces), then the following are equivalent:

1. $T$ is continuous.
2. The graph of $T$ is closed in the product topology on $X\; \backslash times\; Y.$

}}

The usual proof of the closed graph theorem employs the open mapping theorem. It simply uses a general recipe of obtaining the closed graph theorem from the open mapping theorem; see {{section link|closed_graph_theorem|Relation_to_the_open_mapping_theorem}} (this deduction is formal and does not use linearity; the linearity is needed to appeal to the open mapping theorem which relies on the linearity.)

In fact, the open mapping theorem can in turn be deduced from the closed graph theorem as follows. As noted in {{section link|Open_mapping_theorem_(functional_analysis)|Statement_and_proof}}, it is enough to prove the open mapping theorem for a continuous linear operator that is bijective (not just surjective). Let T be such an operator. Then by continuity, the graph $\backslash Gamma\_T$ of T is closed. Then $\backslash Gamma\_T\; \backslash simeq\; \backslash Gamma\_\{T^\{-1\}\}$ under $(x,\; y)\; \backslash mapsto\; (y,\; x)$. Hence, by the closed graph theorem, $T^\{-1\}$ is continuous; i.e., T is an open mapping.

Since the closed graph theorem is equivalent to the open mapping theorem, one knows that the theorem fails without the completeness assumption. But more concretely, an operator with closed graph that is not bounded (see unbounded operator) exists and thus serves as a counterexample.

The Hausdorff–Young inequality says that the Fourier transformation $\backslash widehat\{\backslash cdot\}\; :\; L^p(\backslash mathbb\{R\}^n)\; \backslash to\; L^\{p\text{'}\}(\backslash mathbb\{R\}^n)$ is a well-defined bounded operator with operator norm one when $1/p\; +\; 1/p\text{'}\; =\; 1$. This result is usually proved using the Riesz–Thorin interpolation theorem and is highly nontrivial. The closed graph theorem can be used to prove a soft version of this result; i.e., the Fourier transformation is a bounded operator with the unknown operator norm.{{harvnb|Tao|loc=Example 3}}

Here is how the argument would go. Let T denote the Fourier transformation. First we show $T\; :\; L^p\; \backslash to\; Z$ is a continuous linear operator for Z = the space of tempered distributions on $\backslash mathbb\{R\}^n$. Second, we note that T maps the space of Schwarz functions to itself (in short, because smoothness and rapid decay transform to rapid decay and smoothness, respectively). This implies that the graph of T is contained in $L^p\; \backslash times\; L^\{p\text{'}\}$ and $T\; :\; L^p\; \backslash to\; L^\{p\text{'}\}$ is defined but with unknown bounds.{{clarify|where was the assumption $1/p\; +\; 1/p\text{'}\; =\; 1$ is used?|date=July 2024}} Since $T\; :\; L^p\; \backslash to\; Z$ is continuous, the graph of $T\; :\; L^p\; \backslash to\; L^\{p\text{'}\}$ is closed in the distribution topology; thus in the norm topology. Finally, by the closed graph theorem, $T\; :\; L^p\; \backslash to\; L^\{p\text{'}\}$ is a bounded operator.

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

{{Math theorem|name=Theorem|math_statement=

A linear operator from a barrelled space $X$ to a Fréchet space $Y$ is continuous if and only if its graph is closed.

}}

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

{{Math theorem|name=Theorem|math_statement=

A linear map between two F-spaces is continuous if and only if its graph is closed.{{sfn|Schaefer|Wolff|1999|p=78}}{{harvtxt|Trèves|2006}}, p. 173

}}

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

{{Math theorem|name=Theorem|math_statement=

If $T\; :\; X\; \backslash to\; Y$ is a linear map between two F-spaces, then the following are equivalent:

1. $T$ is continuous.
2. $T$ has a closed graph.
3. If $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}\; \backslash to\; x$ in $X$ and if $T\backslash left(x\_\{\backslash bull\}\backslash right)\; :=\; \backslash left(T\backslash left(x\_i\backslash right)\backslash right)\_\{i=1\}^\{\backslash infty\}$ converges in $Y$ to some $y\; \backslash in\; Y,$ then $y\; =\; T(x).${{sfn|Rudin|1991|pp=50-52}}
4. If $x\_\{\backslash bull\}\; =\; \backslash left(x\_i\backslash right)\_\{i=1\}^\{\backslash infty\}\; \backslash to\; 0$ in $X$ and if $T\backslash left(x\_\{\backslash bull\}\backslash right)$ converges in $Y$ to some $y\; \backslash in\; Y,$ then $y\; =\; 0.$

}}

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

{{Math theorem|name=Closed Graph Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=

Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous.

}}

{{Math theorem|name=Closed Graph Theorem|math_statement=

A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous.{{sfn|Narici|Beckenstein|2011|pp=474-476}}

}}

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|p=479-483}}|math_statement=

Suppose that $T\; :\; X\; \backslash to\; Y$ is a linear map whose graph is closed.

If $X$ is an inductive limit of Baire TVSs and $Y$ is a webbed space then $T$ is continuous.

}}

{{Math theorem|name=Closed Graph Theorem{{sfn|Narici|Beckenstein|2011|pp=474-476}}|math_statement=

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

{{Math theorem|name=Theorem{{sfn|Trèves|2006|p=169}}|math_statement=

Suppose that $X$ and $Y$ are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

:If $G$ is any closed subspace of $X\; \backslash times\; Y$ and $u$ is any continuous map of $G$ onto $X,$ then $u$ is an open mapping.

Under this condition, if $T\; :\; X\; \backslash to\; Y$ is a linear map whose graph is closed then $T$ is continuous.

}}

{{Main|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.{{sfn|Trèves|2006|p=549}}

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:

{{Math theorem|name=Borel Graph Theorem|math_statement=

Let $u\; :\; X\; \backslash to\; Y$ be linear map between two locally convex Hausdorff spaces $X$ and $Y.$ If $X$ is the inductive limit of an arbitrary family of Banach spaces, if $Y$ is a Souslin space, and if the graph of $u$ is a Borel set in $X\; \backslash times\; Y,$ then $u$ is continuous.{{sfn|Trèves|2006|p=549}}

}}

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

A topological space $X$ is called a $K\_\{\backslash sigma\backslash delta\}$ if it is the countable intersection of countable unions of compact sets.

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

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:

{{Math theorem|name=Generalized Borel Graph Theorem{{sfn|Trèves|2006|pp=557-558}}|math_statement=

Let $u\; :\; X\; \backslash to\; Y$ be a linear map between two locally convex Hausdorff spaces $X$ and $Y.$ If $X$ is the inductive limit of an arbitrary family of Banach spaces, if $Y$ is a K-analytic space, and if the graph of $u$ is closed in $X\; \backslash times\; Y,$ then $u$ is continuous.

}}

If $F\; :\; X\; \backslash to\; Y$ is closed linear operator from a Hausdorff locally convex TVS $X$ into a Hausdorff finite-dimensional TVS $Y$ then $F$ is continuous.{{sfn|Narici|Beckenstein|2011|p=476}}

• {{annotated link|Almost open linear map}}
• {{annotated link|Barrelled space}}
• {{annotated link|Closed graph}}
• {{annotated link|Closed linear operator}}
• {{annotated link|Densely defined operator}}
• {{annotated link|Discontinuous linear map}}
• {{annotated link|Kakutani fixed-point theorem}}
• {{annotated link|Open mapping theorem (functional analysis)}}
• {{annotated link|Ursescu theorem}}
• {{annotated link|Webbed space}}

Notes

{{reflist|group=note}}

{{reflist}}

• {{Adasch Topological Vector Spaces}}
• {{Banach Théorie des Opérations Linéaires}}
• {{Berberian Lectures in Functional Analysis and Operator Theory}}
• {{Bourbaki Topological Vector Spaces}}
• {{Conway A Course in Functional Analysis}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Dolecki Mynard Convergence Foundations Of Topology}}
• {{Dubinsky The Structure of Nuclear Fréchet Spaces}}
• {{Grothendieck Topological Vector Spaces}}
• {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}
• {{Jarchow Locally Convex Spaces}}
• {{Köthe Topological Vector Spaces I}}
• {{Kriegl Michor The Convenient Setting of Global Analysis}}
• {{Munkres Topology|edition=2}}
• {{Narici Beckenstein Topological Vector Spaces|edition=2}}
• {{Robertson Topological Vector Spaces}}
• {{Rudin Walter Functional Analysis|edition=2}}
• {{Schaefer Wolff Topological Vector Spaces|edition=2}}
• {{Swartz An Introduction to Functional Analysis}}
• {{citation | last = Tao | first = Terence | author-link = Terence Tao | url = https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/ | title = 245B, Notes 9: The Baire category theorem and its Banach space consequences}}
• {{Trèves François Topological vector spaces, distributions and kernels}}
• {{citation | first = Dietmar | last = Vogt | url = https://www2.math.uni-wuppertal.de/~vogt/vorlesungen/fs.pdf | title = Lectures on Fréchet spaces | date = 2000}}
• {{Wilansky Modern Methods in Topological Vector Spaces}}
• {{planetmath reference|urlname=ProofOfClosedGraphTheorem|title=Proof of closed graph theorem }}

{{Functional Analysis}}

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Category:Theorems in functional analysis