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In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs. Each gives conditions when functions with closed graphs are necessarily continuous. A blog post [1] by T. Tao lists several closed graph theorems throughout mathematics.
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 ...
Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous. If f : X → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous.
Closed graph theorem — If is a topological space and is a compact Hausdorff space, then the graph of a linear map from to is closed if and only if is continuous. [ 9 ] Other topics
In functional analysis, a branch of mathematics, a closed linear operator or often a closed operator is a linear operator whose graph is closed (see closed graph property). It is a basic example of an unbounded operator. The closed graph theorem says a linear operator between Banach spaces is a closed operator if and only if it is a bounded ...
Theorem — If : is an upper hemicontinuous set-valued function with closed domain (that is, the domain of is closed) and closed values (i.e. () is closed for all ), then is closed. If B {\displaystyle B} is compact, then the converse is also true.
Closed graph theorem (functional analysis) Closed range theorem (functional analysis) Cluster decomposition theorem (quantum field theory) Coase theorem ; Cochran's theorem ; Codd's theorem (relational model) Cohen structure theorem (commutative algebra) Cohn's irreducibility criterion (polynomials)
By the closed graph theorem, is in the spectrum if and only if the bounded operator : is non-bijective on . The study of spectra and related properties is known as spectral theory , which has numerous applications, most notably the mathematical formulation of quantum mechanics .