Closed graph theorem (functional analysis)

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 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. 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. If the closed graph theorem applies, then T is continuous under the original topology. See for an explicit example. ==Statement ==

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 denote 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 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 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. == Generalization ==

On Hilbert spaces Let H be a Hilbert space, and A: D(A) \subset H \to H be a possibly partially-defined linear operator. Define the graph inner product on D(A) by \langle x, y\rangle_{G(A)}=\langle x, y\rangle+\langle A x, A y\rangle, and similarly the graph norm \|x \|_{G(A)} = \sqrt{\|x\|^2 + \|Ax\|^2}. We have the following: • A is closed iff (D(A), \|\cdot \|_{G(A)}) is Banach. • If A is bounded, then A is closed iff 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. Between F-spaces There are versions that does not require 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: {{Math theorem|name=Theorem|math_statement= If T : X \to Y is a linear map between two F-spaces, then the following are equivalent: • T is continuous. • T has a closed graph. • If x_{\bull} = \left(x_i\right)_{i=1}^{\infty} \to x in X and if T\left(x_{\bull}\right) := \left(T\left(x_i\right)\right)_{i=1}^{\infty} converges in Y to some y \in Y, then y = T(x). • If x_{\bull} = \left(x_i\right)_{i=1}^{\infty} \to 0 in X and if T\left(x_{\bull}\right) converges in Y to some y \in Y, then y = 0. }} 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. 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 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: ==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. ==See also==

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