Closed graph theorem

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

Formulation

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

Then is bounded if (and therefore continuous) when a closed operator (that is completed ).

Derivation

Because of the seclusion of the graph is a Banach space. Trivially is a bijective bounded linear mapping between and. From the set of the open mapping then follows that the converse is also limited.

Generalization

The set of closed graph can be extended in the theory of locally convex spaces to larger room classes, see space with tissue ultrabornologischer space or (LF )-space.

Application

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