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The closed graph theorem is an important result in functional analysis that guarantees that a closed linear operator is continuous under certain conditions. The original result has been generalized many times. A well known version of the closed graph theorems is the following.
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.
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 closed graph theorem § Relation to the open mapping theorem (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 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.
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 .
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)
Closed Graph Theorem [6] — Let : be a linear map between TVSs that is sequentially closed (meaning that its graph is a sequentially closed subset of ). If Y {\displaystyle Y} is a webbed space and X {\displaystyle X} is an ultrabornological space (such as a Fréchet space or an inductive limit of Fréchet spaces), then A {\displaystyle A} is ...