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In mathematics, the closed graph theorem is a basic result in functional analysis which characterizes continuous linear operators between Banach spaces in terms of the operator graph. Euclid, detail from The School of Athens by Raphael. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
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. ...
In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
The closed graph theorem
For any function T : X → Y, we define the graph of T to be the set { (x,y) ∈ X×Y | y = T(x) }.
Suppose that X and Y are Banach spaces, and that T is an everywhere-defined linear operator (i.e. the domain D(T) of T is X). Then T is continuous if and only if it is a closed operator, that is, its graph is closed in X×Y (with the product topology). The restriction on the domain is needed due to the existence of closed unbounded linear operators. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
In mathematics, the domain of a function is the set of all input values to the function. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...
In mathematics, specifically in functional analysis, closed linear operators are an important class of linear operators on Banach spaces. ...
In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
The usual proof of the closed graph theorem employs the open mapping theorem. In mathematics, there are two theorems with the name 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: if A : X → Y is a surjective continuous linear operator between Banach spaces X and Y...
The closed graph theorem can be reformulated as follows. If T : X → Y is a linear operator between Banach spaces, then the following are equivalent:
- If the sequence {xn} in X converges to some element x, then the sequence {T(xn)} in Y also converges, and its limit is T(x).
- If the sequence {xn} in X converges to some element x and the {T(xn)} in Y converges to some element y, then y = T(x).
Generalization The closed graph theorem can be generalized to more abstract topological vector spaces in the following way: In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard...
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. In functional analysis and related areas of mathematics barrelled spaces are topological vector spaces where every barrelled set in the space is a neighbourhood for the zero vector. ...
This article deals with Fréchet spaces in functional analysis. ...
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. ...
See also In mathematics, linear maps form an important class of simple functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). ...
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