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.
Since an operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator, one can replace "continuous" with "bounded" in the statement above.
An important question in functional analysis is whether a given linear operator is continuous. The closed graph theorem gives one answer to that question.
Explanation
Let be a linear operator between Banach spaces. Then the continuity of means that for each convergent sequence. On the other hand, the closedness of the graph of means that for each convergent sequence such that, we have. Hence, the closed graph theorem says that in order to check the continuity of, one can show under the additional assumption that is convergent.In fact, for the graph of T to be closed, it is enough that if, then. Indeed, assuming that condition holds, if, then and. Thus, ; i.e., 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; seeIn 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. Let T be such an operator. Then by continuity, the graph of T is closed. Then under. Hence, by the closed graph theorem, 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 exists and thus serves as a counterexample.
Example
The Hausdorff–Young inequality says that the Fourier transformation is a well-defined bounded operator with operator norm one when. 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 is a continuous linear operator for Z = the space of tempered distributions on. Second, we note that T maps the space of Schwartz functions to itself. This implies that the graph of T is contained in and is defined but with unknown bounds. Since is continuous, the graph of is closed in the distribution topology; thus in the norm topology. Finally, by the closed graph theorem, is a bounded operator.
Generalization
On Hilbert spaces
Let be a Hilbert space, and be a possibly partially-defined linear operator.Define the graph inner product on by, and similarly the graph norm. We have the following:
- is closed iff is Banach.
- If is bounded, then is closed iff is closed.
Complete metrizable codomain
Between F-spaces
There are versions that does not require 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 isBorel 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 is called a if it is the countable intersection of countable unions of compact sets.
A Hausdorff topological space is called K-analytic if it is the continuous image of a space.
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: