Open mapping theorem (functional analysis)


In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem, is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map.
More precisely, :

Open mapping theorem for Banach spaces

One proof uses Baire's category theorem, and completeness of both and is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if and are taken to be Fréchet spaces.

Related results

Consequences

The open mapping theorem has several important consequences:
Local convexity of  or  is not essential to the proof, but completeness is: the theorem remains true in the case when and are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner:
Furthermore, in this latter case if is the kernel of, then there is a canonical factorization of in the form
where is the quotient space of by the closed subspace.
The quotient mapping is open, and the mapping α is an isomorphism of topological vector spaces.
The open mapping theorem can also be stated as

Consequences

Webbed spaces

s are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.