In functional analysis, the open mapping theorem, also known as the BanachâÂÂSchauder theorem or the Banach theorem (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.
A special case is also called the bounded inverse theorem (also called inverse mapping theorem or Banach isomorphism theorem), which states that a bijective bounded linear operator from one Banach space to another has bounded inverse .
The proof here uses the Baire 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 assumed to be only a normed vector space; see .
The proof is based on the following lemmas, which are also somewhat of independent interest. A linear map between topological vector spaces is said to be nearly open if, for each neighborhood of zero, the closure contains a neighborhood of zero. The next lemma may be thought of as a weak version of the open mapping theorem.
Proof: Shrinking , we can assume is an open ball centered at zero. We have . Thus, some contains an interior point ; that is, for some radius ,
Then for any in with , by linearity, convexity and ,
which proves the lemma by dividing by . (The same proof works if are pre-Fréchet spaces.)
The completeness on the domain then allows to upgrade nearly open to open.
Proof: Let be in and some sequence. We have: . Thus, for each and in , we can find an with and in . Thus, taking , we find an such that
Applying the same argument with , we then find an such that
where we observed . Then so on. Thus, if , we found a sequence such that converges and . Also,
Since , by making small enough, we can achieve . (Again the same proof is valid if are pre-Fréchet spaces.)
Proof of the theorem: By Baire's category theorem, the first lemma applies. Then the conclusion of the theorem follows from the second lemma.
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:
Even though the above bounded inverse theorem is a special case of the open mapping theorem, the open mapping theorem in turn follows from that. Indeed, a surjective continuous linear operator factors as
Here, 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, is open then.
Because the open mapping theorem and the bounded inverse theorem are essentially the same result, they are often simply called Banach's theorem.
Here is a formulation of the open mapping theorem in terms of the transpose of an operator.
Proof: The idea of 1. 2. is to show: and that follows from the HahnâÂÂBanach theorem. 2. 3. is exactly the second lemma in . Finally, 3. 4. is trivial and 4. 1. easily follows from the open mapping theorem.
Alternatively, 1. implies that is injective and has closed image and then by the closed range theorem, that implies has dense image and closed image, respectively; i.e., is surjective. Hence, the above result is a variant of a special case of the closed range theorem.
Terence Tao gives the following quantitative formulation of the theorem:
The proof follows a cycle of implications . Here is the usual open mapping theorem.
Finally, is trivial.
The open mapping theorem may not hold for normed spaces that are not complete. A quickest way to see this is to note that the closed graph theorem, a consequence of the open mapping theorem, fails without completeness. But here is a more concrete counterexample. Consider the space of sequences with only finitely many non-zero terms equipped with the supremum norm. The map defined by
is bounded, linear and invertible, but is unbounded. This does not contradict the bounded inverse theorem since is not complete, and thus is not a Banach space. To see that it's not complete, consider the sequence of sequences given by
converges as to the sequence given by
which has all its terms non-zero, and so does not lie in .
The completion of is the space of all sequences that converge to zero, which is a (closed) subspace of the âÂÂ<sup>p</sup> space , which is the space of all bounded sequences. However, in this case, the map is not onto, and thus not a bijection. To see this, one need simply note that the sequence
is an element of , but is not in the range of . The same reasoning applies to show is also not onto in , for example is not in the range of .
Even if the domain is complete (or the codomain is), the Open Mapping Theorem still requires both spaces to be complete. To see this, consider the identity map from the space of absolutely summable sequences (that is, those with finite 1-norm) with the 1-norm to the space with the supremum norm. Since this map is norm decreasing, it is bounded, but it is not open. To see that the domain must also be complete, let be a Banach space with a discontinuous linear functional on it. Then is a non-complete normed space, and the identity map from to is a norm-decreasing (therefore bounded) map which is not open.
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. What we have is:
In particular, the above applies to an operator between Hilbert spaces or an operator with finite-dimensional kernel (by the HahnâÂÂBanach theorem). 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 BartleâÂÂGraves theorem.
Local convexity of or âÂÂis 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:
(The proof is essentially the same as the Banach or Fréchet cases; we modify the proof slightly to avoid the use of convexity,)
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 (also an F-space) of by the closed subspace The quotient mapping is open, and the mapping is an isomorphism of topological vector spaces.
An important special case of this theorem can also be stated as
On the other hand, a more general formulation, which implies the first, can be given:
Nearly/Almost open linear maps
A linear map between two topological vector spaces (TVSs) is called a (or sometimes, an ) if for every neighborhood of the origin in the domain, the closure of its image is a neighborhood of the origin in Many authors use a different definition of "nearly/almost open map" that requires that the closure of be a neighborhood of the origin in rather than in but for surjective maps these definitions are equivalent. A bijective linear map is nearly open if and only if its inverse is continuous. Every surjective linear map from locally convex TVS onto a barrelled TVS is nearly open. The same is true of every surjective linear map from a TVS onto a Baire TVS.
Webbed spaces are a class of topological vector spaces for which the open mapping theorem and the closed graph theorem hold.