Open mapping theorem (functional analysis)

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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, (Rudin 1966, Theorem 2.11):

If X and Y are Banach spaces and A : X → Y is a surjective continuous linear operator, then A is an open map (i.e. if U is an open set in X, then A(U) is open in Y).

The proof uses the Baire category theorem, and completeness of both X and Y 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 X and Y are taken to be Fréchet spaces.

[edit] Consequences

The open mapping theorem has several important consequences:

If A : X → Y is a bijective continuous linear operator between the Banach spaces X and Y, then the inverse operator A^-1 : Y → X is continuous as well (this is called the bounded inverse theorem). (Rudin 1966, Corollary 2.12)
If A : X → Y is a linear operator between the Banach spaces X and Y, and if for every sequence (x_n) in X with x_n → 0 and Ax_n → y it follows that y = 0, then A is continuous (Closed graph theorem). (Rudin 1966, Theorem 2.15)

[edit] Generalizations

Although X and Y must be complete, the theorem remains true in the case when X and Y are F-spaces. Furthermore, the theorem can be combined with the Baire category theorem in the following manner (Rudin, Theorem 2.11):

Let X be a F-space and Y a topological vector space. If A:X → Y is a continuous linear operator, then either A(X) is a meager set in Y, or A(X) = Y. In the latter case, A is an open mapping and Y is also an F-space.

Furthermore, in this latter case if N is the kernel of A, then there is a canonical factorization of A in the form

$X\to X/N \overset{\alpha}{\to} Y$

where X/N is the quotient space (also an F-space) of X by the closed subspace N. The quotient mapping X → X/N is open, and the mapping α is an isomorphism of topological vector spaces (Diedonné, 12.16.8).

[edit] References

Rudin, Walter (1966), Real & Complex Analysis, McGraw-Hill, ISBN 0-07-054234-1
Dieudonné, Jean (1970), Treatise on Analysis, Volume II, Academic Press