Closed graph theorem (functional analysis)

In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that a linear operator between two Banach spaces is continuous if and only if the graph of the operator is closed (such an operator is called a closed linear operator; see also closed graph property).

One of important questions in functional analysis is the question of the continuity (or boundedness) of a given linear operator. The closed graph theorem gives one answer to that question.

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

In fact, for the graph of T to be closed, it is enough that if $$x_i \to 0, \, Tx_i \to y$$, then $$y = 0$$. Indeed, assuming that condition holds, if $$(x_i, Tx_i) \to (x, y)$$, then $$x_i - x \to 0$$ and $$T(x_i - x) \to y - Tx$$. 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 necessarily 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. 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 for an explicit example.

Statement
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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 (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, 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 $$\Gamma_T$$ of T is closed. Then $$\Gamma_T \simeq \Gamma_{T^{-1}}$$ under $$(x, y) \mapsto (y, x)$$. 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.

Example
The Hausdorff–Young inequality says that the Fourier transformation $$\widehat{\cdot} : L^p(\mathbb{R}^n) \to L^{p'}(\mathbb{R}^n)$$ is a well-defined bounded operator with operator norm one when $$1/p + 1/p' = 1$$. 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 denotes the Fourier transformation. First we show $$T : L^p \to Z$$ is a continuous linear operator for Z = the space of tempered distributions on $$ \mathbb{R}^n$$. 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 $$L^p \times L^{p'}$$ and $$T : L^p \to L^{p'}$$ is defined but with unknown bounds. Since $$T : L^p \to Z$$ is continuous, the graph of $$T : L^p \to L^{p'}$$ is closed in the distribution topology; thus in the norm topology. Finally, by the closed graph theorem, $$T : L^p \to L^{p'}$$ is a bounded operator.

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

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Between F-spaces
There are versions that does not require $$Y$$ to be locally convex.

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

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

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Codomain not complete or (pseudo) metrizable
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An even more general version of the closed graph theorem is

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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:

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

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

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Related results
If $$F : X \to Y$$ is closed linear operator from a Hausdorff locally convex TVS $$X$$ into a Hausdorff finite-dimensional TVS $$Y$$ then $$F$$ is continuous.