# Open mapping theorem (functional analysis)

In functional analysis, the **open mapping theorem**, also known as the **Banach–Schauder theorem** (named after Stefan Banach and Juliusz Schauder), 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 1973, Theorem 2.11):

**Open Mapping Theorem.**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*).

One proof uses Baire's 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.

## 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 1973, Corollary 2.12) - If
*A*:*X*→*Y*is a linear operator between the Banach spaces*X*and*Y*, and if for every sequence (*x*) in_{n}*X*with*x*→ 0 and_{n}*Ax*→_{n}*y*it follows that*y*= 0, then*A*is continuous (the closed graph theorem). (Rudin 1973, Theorem 2.15)

## Proof

Suppose *A* : *X* → *Y* is a surjective continuous linear operator. In order to prove that *A* is an open map, it is sufficient to show that *A* maps the open unit ball in *X* to a neighborhood of the origin of *Y*.

Let . Then

- .

Since *A* is surjective:

But *Y* is Banach so by Baire's category theorem

That is, we have *c* in *Y* and *r* > 0 such that

Let *v* ∈ *V*, then

By continuity of addition and linearity, the difference *rv* satisfies

and by linearity again,

where we have set *L*=2*k*/*r*. It follows that

Our next goal is to show that *V* ⊆ *A*(2*LU*).

Let *y* ∈ *V*. By (1), there is some *x*_{1} with ||*x*_{1}|| < L and ||*y* − *Ax*_{1}|| < 1/2. Define a sequence {*x _{n}*} inductively as follows. Assume:

Then by (1) we can pick *x*_{n+1} so that:

so (2) is satisfied for *x*_{n+1}. Let

From the first inequality in (2), {*s _{n}*} is a Cauchy sequence, and since

*X*is complete,

*s*converges to some

_{n}*x*∈

*X*. By (2), the sequence

*As*tends to

_{n}*y*, and so

*Ax*=

*y*by continuity of

*A*. Also,

This shows that *y* belongs to *A*(2*LU*), so *V* ⊆ *A*(2*LU*) as claimed. Thus the image *A*(*U*) of the unit ball in *X* contains the open ball *V*/2*L* of *Y*. Hence, *A*(*U*) is a neighborhood of 0 in *Y*, and this concludes the proof.

## Generalizations

Local convexity of *X* or *Y* is not essential to the proof, but completeness is: 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

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 (Dieudonné, 12.16.8).

The open mapping theorem can also be stated as[1]

- Let
*X*and*Y*be two F-spaces. Then every continuous linear map of*X*onto*Y*is a TVS homomorphism.

where a linear map *u* : *X* → *Y* is a topological vector space (TVS) homomorphism if the induced map is a TVS-isomorphism onto its image.

## References

- Trèves (1967), p. 170

- Rudin, Walter (1973),
*Functional Analysis*, McGraw-Hill, ISBN 0-07-054236-8 - Dieudonné, Jean (1970),
*Treatise on Analysis, Volume II*, Academic Press - Trèves, François (1967),
*Topological Vector Spaces, Distributions and Kernels*, Academic Press, Inc., pp. 166, 170, ISBN 0-486-45352-9

*This article incorporates material from Proof of open mapping theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*