Open mapping theorem (functional analysis)

The set of the open mapping, also known as the Banach - Schauder, is a basic set of functional analysis, a branch of mathematics. The theorem is a consequence of the theorem of Baire and was proved in 1929 by Stefan Banach, Juliusz Schauder and.

Statement

A map between topological spaces is called open if the image of every open set is open.

The statement of the theorem is:

It is easily seen that an open linear map must be surjective, there is no true subspace of being open; the content of the sentence thus lies in the statement that every surjective continuous linear map is open. The proof requires both the completeness of, and the of.

Set of continuous inverse

Immediately from the definition of continuity follows as a corollary:

This statement is known as the law of the inverse mapping or set of continuous inverse. It can also be formulated as follows:

Generalization

The theorem on the open mapping can be extended in the theory of locally convex spaces to larger room classes, see space with tissue ultrabornologischer space or (LF )-space.