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.
The closed graph theorem gives one answer to that question.
Hence, the closed graph theorem says that in order to check the continuity of
[1] 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).
is a linear operator between Banach spaces (or more generally Fréchet spaces), then the following are equivalent: 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.)
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).
But more concretely, an operator with closed graph that is not bounded (see unbounded operator) exists and thus serves as a counterexample.
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.
is a continuous linear operator for Z = the space of tempered distributions on
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).
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
Theorem — A linear map between two F-spaces is continuous if and only if its graph is closed.
is a linear map between two F-spaces, then the following are equivalent: Every metrizable topological space is pseudometrizable.
Closed Graph Theorem[7] — 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.
Closed Graph Theorem[7] — A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous.
are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property: Under this condition, if
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.
be linear map between two locally convex Hausdorff spaces
is the inductive limit of an arbitrary family of Banach spaces, if
[10] An improvement upon this theorem, proved by A. Martineau, uses 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[11] — Let
be a linear map between two locally convex Hausdorff spaces
is the inductive limit of an arbitrary family of Banach spaces, if
is closed linear operator from a Hausdorff locally convex TVS