In functional analysis, the open mapping theorem, also known as the Banach–Schauder 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.

Statement and proof

The proof here uses the Baire category theorem, and completeness 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 .

between topological vector spaces is said to be nearly open contains a neighborhood of zero. The next lemma may be thought of as a weak version of the open mapping theorem.

,

,

1. ,

are pre-Fréchet spaces.)

The completeness on the domain then allows to upgrade nearly open to open.

such that

such that

. Also,

are pre-Fréchet 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:

factors as

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.

Transpose formulation

Here is a formulation of the open mapping theorem in terms of the transpose of an operator.

and that follows from the Hahn–Banach theorem 3. is exactly the second lemma in

is injective and has closed image and then by the closed range theorem is surjective. Hence, the above result is a variant of a special case of the closed range theorem.

Quantitative formulation

Terence Tao gives the following quantitative formulation of the theorem:

The proof follows a cycle of implications is the usual open mapping theorem.

.

is a required solution.

Counterexample

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 of sequences with only finitely many non-zero terms equipped with the supremum norm defined by

is not complete given by

given by

.

is the c_0 of all sequences that converge to zero, which is a (closed) subspace of the ℓp space is not onto, and thus not a bijection. To see this, one need simply note that the sequence

.

is a norm-decreasing (therefore bounded) map which is not open.

Consequences

The open mapping theorem has several important consequences:

• is a bijective then the inverse operator is continuous as well (this is called the bounded inverse theorem).

• and if for every sequence is continuous (the closed graph theorem).

• is non-meager is complete (to see this, use the two lemmas in the proof of the theorem).

• An exact sequence of Banach spaces (or more generally Fréchet spaces) is topologically exact.

• The closed range theorem, which says an operator (under some assumption) has closed image if and only if its transpose has closed image (see closed range theorem#Sketch of proof).

The open mapping theorem does not imply that a continuous surjective linear operator admits a continuous linear section. What we have is:

• A surjective continuous linear operator between Banach spaces admits a continuous linear section if and only if the kernel is topologically complemented.

In particular, the above applies to an operator between Hilbert spaces or an operator with finite-dimensional kernel (by the Hahn–Banach 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 Bartle–Graves theorem.

Generalizations

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,)

is the kernel is the quotient space by the closed 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

between two topological vector spaces (TVSs) is called a (or sometimes, an 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.

See also

References

Bibliography

Further reading