Closed graph theorem
Encyclopedia
In mathematics
, the closed graph theorem is a basic result in functional analysis
which characterizes continuous linear operator
s between Banach space
s in terms of the operator graph
.
If X is any topological space
and Y is Hausdorff
, then it is straightforward to show that the graph of T is closed whenever T is continuous.
If X and Y are Banach spaces, and T is an everywhere-defined (i.e. the domain
D(T) of T is X) linear operator, then the converse is true as well. This is the content of the closed graph theorem: if the graph of T is closed
in (with the product topology
), we say that T is a closed operator
, and, in this setting, we may conclude that T is continuous.
The restriction on the domain is needed due to the existence of closed unbounded
linear operators. The differentiation operator on
is a prototypical counter-example.
The usual proof of the closed graph theorem employs the open mapping theorem
. In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem
are all equivalent. This equivalence also serves to demonstrate the necessity of X and Y being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.
The closed graph theorem can be reformulated as follows. If is a linear bounded operator between Banach spaces, then the following are equivalent:
A linear operator from a barrelled space
X to a Fréchet space
Y is continuous
if and only if its graph is closed in the space X×Y equipped with the product topology.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the closed graph theorem is a basic result in functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
which characterizes continuous linear operator
Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
s between Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s in terms of the operator graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
.
The closed graph theorem
For any function , we define the graph of T to be the setIf X is any topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and Y is Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
, then it is straightforward to show that the graph of T is closed whenever T is continuous.
If X and Y are Banach spaces, and T is an everywhere-defined (i.e. the domain
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
D(T) of T is X) linear operator, then the converse is true as well. This is the content of the closed graph theorem: if the graph of T is closed
Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
in (with the product topology
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
), we say that T is a closed operator
Closed operator
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the...
, and, in this setting, we may conclude that T is continuous.
The restriction on the domain is needed due to the existence of closed unbounded
Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases....
linear operators. The differentiation operator on
is a prototypical counter-example.The usual proof of the closed graph theorem employs the open mapping theorem
Open mapping theorem (functional analysis)
In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem , is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map...
. In fact, the closed graph theorem, the open mapping theorem and the bounded inverse theorem
Bounded inverse theorem
In mathematics, the bounded inverse theorem is a result in the theory of bounded linear operators on Banach spaces. It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both the open mapping theorem and the closed...
are all equivalent. This equivalence also serves to demonstrate the necessity of X and Y being Banach; one can construct linear maps that have unbounded inverses in this setting, for example, by using either continuous functions with compact support or by using sequences with finitely many non-zero terms along with the supremum norm.
The closed graph theorem can be reformulated as follows. If is a linear bounded operator between Banach spaces, then the following are equivalent:
- For every sequence {xn} in X, if the sequence {xn} converges in X to some element x, then the sequence {T(xn)} in Y also converges, and its limit is T(x).
- For every sequence {xn} in X, if the sequence {xn} converges in X to some element x and the sequence {T(xn)} in Y converges to some element y, then .
Generalization
The closed graph theorem can be generalized to more abstract topological vector spaces in the following way:A linear operator from a barrelled space
Barrelled space
In functional analysis and related areas of mathematics, barrelled spaces are Hausdorff topological vector spaces for which every barrelled set in the space is a neighbourhood for the zero vector. A barrelled set or a barrel in a topological vector space is a set which is convex, balanced,...
X to a Fréchet space
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
Y is continuous
Continuous linear operator
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces....
if and only if its graph is closed in the space X×Y equipped with the product topology.