Theorem of the closed graph

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The theorem of the closed graph is a mathematical theorem from functional analysis .

formulation

Let and be Banach spaces and a linear operator . Let it denote the graph of .

Then is bounded (and thus continuous) if and only if is a closed operator (i.e. closed in ).

Derivation

The theorem of the closed graph can be reduced to Zabreiko's lemma .

Furthermore, the sentence can be derived from the sentence of the open mapping as follows . Because the graph is closed, it is a Banach space. Trivially, is a bijective, linear, bounded mapping between and . It then follows from the theorem of the open mapping that the inversion is also bounded, and that implies the continuity of .

generalization

In the theory of locally convex spaces, the theorem of the closed graph can be extended to larger space classes, see space with tissue , ultrabornological space or (LF) space .

application

The set of Hellinger-Toeplitz is a consequence of the principle of closed graph.

literature