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.
Let
T:X\toY
T
Txi\toTx
xi\tox
T
xi\tox
Txi\toy
y=Tx
T
Txi\toTx
Txi
In fact, for the graph of T to be closed, it is enough that if
xi\to0,Txi\toy
y=0
(xi,Txi)\to(x,y)
xi-x\to0
T(xi-x)\toy-Tx
y=Tx
(x,y)
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.[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). See for an explicit example.
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
\GammaT
\GammaT\simeq
\Gamma | |
T-1 |
(x,y)\mapsto(y,x)
T-1
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.
The Hausdorff–Young inequality says that the Fourier transformation
\widehat{ ⋅ }:Lp(Rn)\toLp'(Rn)
1/p+1/p'=1
Here is how the argument would go. Let T denote the Fourier transformation. First we show
T:Lp\toZ
Rn
Lp x Lp'
T:Lp\toLp'
T:Lp\toZ
T:Lp\toLp'
T:Lp\toLp'
The closed graph theorem can be generalized from Banach spaces to more abstract topological vector spaces in the following ways.
There are versions that does not require
Y
This theorem is restated and extend it with some conditions that can be used to determine if a graph is closed:
Every metrizable topological space is pseudometrizable. A pseudometrizable space is metrizable if and only if it is Hausdorff.
An even more general version of the closed graph theorem is
See main article: 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
K\sigma\delta
A Hausdorff topological space
Y
K\sigma\delta
K\sigma\delta
X
X
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:
If
F:X\toY
X
Y
F
Notes