In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem or the Banach theorem[1] (named after Stefan Banach and Juliusz Schauder), is a fundamental result that states that if a bounded or continuous linear operator between Banach spaces is surjective then it is an open map.
The proof here uses the Baire category theorem, and completeness of both
The proof is based on the following lemmas, which are also somewhat of independent interest.
between topological vector spaces is said to be nearly open if, for each neighborhood
The next lemma may be thought of as a weak version of the open mapping theorem.
be a continuous linear map between normed spaces.
In general, a continuous bijection between topological spaces is not necessarily a homeomorphism.
The open mapping theorem, when it applies, implies the bijectivity is enough: Corollary (Bounded inverse theorem) — [8] A continuous bijective linear operator between Banach spaces (or Fréchet spaces) has continuous inverse.
is continuous and bijective and thus is a homeomorphism by the bounded inverse theorem; in particular, it is an open mapping.
As a quotient map for topological groups is open,
Here is a formulation of the open mapping theorem in terms of the transpose of an operator.
Hence, the above result is a variant of a special case of the closed range theorem.
in the dense subspace and the sum converging in norm.
The open mapping theorem may not hold for normed spaces that are not complete.
Consider the space X of sequences x : N → R with only finitely many non-zero terms equipped with the supremum norm.
The map T : X → X defined by is bounded, linear and invertible, but T−1 is unbounded.
This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space.
To see this, one need simply note that the sequence is an element of
The open mapping theorem has several important consequences: The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section.
If one drops the requirement that a section be linear, a surjective continuous linear operator between Banach spaces admits a continuous section; this is the Bartle–Graves theorem.
is not essential to the proof, but completeness is: the theorem remains true in the case when
be a continuous linear operator from a complete pseudometrizable TVS
is a topological vector space (TVS) homomorphism if the induced map
On the other hand, a more general formulation, which implies the first, can be given: Open mapping theorem[15] — Let
be a surjective linear map from a complete pseudometrizable TVS
is (a closed linear operator and thus also) an open mapping.
[18] Many authors use a different definition of "nearly/almost open map" that requires that the closure of
A bijective linear map is nearly open if and only if its inverse is continuous.
is a continuous linear bijection from a complete Pseudometrizable topological vector space (TVS) onto a Hausdorff TVS that is a Baire space, then
Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.