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Concept
Closed graph property
Summary
In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.
A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y.
A related property is open graph.
This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.
Definition and notation: The graph of a function f : X → Y is the set
Gr f := { (x, f(x)) : x ∈ X } = { (x, y) ∈ X × Y : y = f(x) }.
Notation: If Y is a set then the power set of Y, which is the set of all subsets of Y, is denoted by 2Y or P(Y).
Definition: If X and Y are sets, a set-valued function in Y on X (also called a Y-valued multifunction on X) is a function F : X → 2Y with domain X that is valued in 2Y. That is, F is a function on X such that for every x ∈ X, F(x) is a subset of Y.
Some authors call a function F : X → 2Y a set-valued function only if it satisfies the additional requirement that F(x) is not empty for every x ∈ X; this article does not require this.
Definition and notation: If F : X → 2Y is a set-valued function in a set Y then the graph of F is the set
Gr F := { (x, y) ∈ X × Y : y ∈ F(x) }.
Definition: A function f : X → Y can be canonically identified with the set-valued function F : X → 2Y defined by F(x) := { f(x) } for every x ∈ X, where F is called the canonical set-valued function induced by (or associated with) f.
Note that in this case, Gr f = Gr F.
We give the more general definition of when a Y-valued function or set-valued function defined on a subset S of X has a closed graph since this generality is needed in the study of closed linear operators that are defined on a dense subspace S of a topological vector space X (and not necessarily defined on all of X).
This particular case is one of the main reasons why functions with closed graphs are studied in functional analysis.
Official source
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