Closed graph theorem (functional analysis)

Statement

Theorem— If ${\displaystyle T:X\to Y}$ is a linear operator between Banach spaces (or more generally Fréchet spaces), then the following are equivalent:

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

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 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 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 ${\displaystyle {\mathrm{\Gamma }}_T}$ of T is closed. Then ${\displaystyle {\mathrm{\Gamma }}_T\simeq {\mathrm{\Gamma }}_{T^{-1}}}$ under ${\displaystyle (x,y)\mapsto (y,x)}$. Hence, by the closed graph theorem, ${\displaystyle T^{-1}}$ 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.

Example

The Hausdorff–Young inequality says that the Fourier transformation ${\displaystyle \hat{\cdot }:L^p({\mathbb{R}}^n)\to L^{p^{\prime }}({\mathbb{R}}^n)}$ is a well-defined bounded operator with operator norm one when ${\displaystyle 1/p+1/p^{\prime }=1}$. 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.

Here is how the argument would go. Let T denote the Fourier transformation. First we show ${\displaystyle T:L^p\to Z}$ is a continuous linear operator for Z = the space of tempered distributions on ${\displaystyle {\mathbb{R}}^n}$. Second, we note that T maps the space of Schwartz 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 ${\displaystyle L^p\times L^{p^{\prime }}}$ and ${\displaystyle T:L^p\to L^{p^{\prime }}}$ is defined but with unknown bounds. Since ${\displaystyle T:L^p\to Z}$ is continuous, the graph of ${\displaystyle T:L^p\to L^{p^{\prime }}}$ is closed in the distribution topology; thus in the norm topology. Finally, by the closed graph theorem, ${\displaystyle T:L^p\to L^{p^{\prime }}}$ is a bounded operator.

Generalization

On Hilbert spaces

Let ${\displaystyle H}$ be a Hilbert space, and ${\displaystyle A:D(A)\subset H\to H}$ be a possibly partially-defined linear operator.

Define the graph inner product on ${\displaystyle D(A)}$ by ${\displaystyle ⟨x,y{⟩}_{G(A)}=⟨x,y⟩+⟨Ax,Ay⟩}$, and similarly the graph norm ${\displaystyle \Vert x{\Vert }_{G(A)}=\sqrt{\Vert x{\Vert }^2+\Vert Ax{\Vert }^2}}$. We have the following:

• ${\displaystyle A}$ is closed iff ${\displaystyle (D(A),\Vert \cdot {\Vert }_{G(A)})}$ is Banach.
• If ${\displaystyle A}$ is bounded, then ${\displaystyle A}$ is closed iff ${\displaystyle D(A)}$ is closed.

Complete metrizable codomain

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 ${\displaystyle X}$ to a Fréchet space ${\displaystyle Y}$ is continuous if and only if its graph is closed.

Between F-spaces

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

Theorem—A linear map between two F-spaces is continuous if and only if its graph is closed.

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

Theorem—If ${\displaystyle T:X\to Y}$ is a linear map between two F-spaces, then the following are equivalent:

1. ${\displaystyle T}$ is continuous.
2. ${\displaystyle T}$ has a closed graph.
3. If ${\displaystyle x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}\to x}$ in ${\displaystyle X}$ and if ${\displaystyle T\left(x_{\bullet }\right):={\left(T\left(x_i\right)\right)}_{i=1}^{\mathrm{\infty }}}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y,}$ then ${\displaystyle y=T(x).}$
4. If ${\displaystyle x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}\to 0}$ in ${\displaystyle X}$ and if ${\displaystyle T\left(x_{\bullet }\right)}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y,}$ then ${\displaystyle y=0.}$

Complete pseudometrizable codomain

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

Closed Graph Theorem—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.

Codomain not complete or (pseudo) metrizable

Theorem—Suppose that ${\displaystyle T:X\to Y}$ is a linear map whose graph is closed. If ${\displaystyle X}$ is an inductive limit of Baire TVSs and ${\displaystyle Y}$ is a webbed space then ${\displaystyle T}$ is continuous.

Closed Graph Theorem—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—Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

If ${\displaystyle G}$ is any closed subspace of ${\displaystyle X\times Y}$ and ${\displaystyle u}$ is any continuous map of ${\displaystyle G}$ onto ${\displaystyle X,}$ then ${\displaystyle u}$ is an open mapping.

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

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. 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 ${\displaystyle u:X\to Y}$ be linear map between two locally convex Hausdorff spaces ${\displaystyle X}$ and ${\displaystyle Y.}$ If ${\displaystyle X}$ is the inductive limit of an arbitrary family of Banach spaces, if ${\displaystyle Y}$ is a Souslin space, and if the graph of ${\displaystyle u}$ is a Borel set in ${\displaystyle X\times Y,}$ then ${\displaystyle u}$ is continuous.

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

A topological space ${\displaystyle X}$ is called a ${\displaystyle K_{\sigma \delta }}$ 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 ${\displaystyle K_{\sigma \delta }}$ space (that is, if there is a ${\displaystyle K_{\sigma \delta }}$ 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:

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

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 ${\displaystyle F}$ is continuous.

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