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Open mapping theorem (functional analysis)

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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):

Contents

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 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.

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 (xn) in X with xn → 0 and Axn → 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 U = B 1 X ( 0 ) , V = B 1 Y ( 0 ) . Then

    X = ⋃ k ∈ N k U .

    Since A is surjective:

    Y = A ( X ) = A ( ⋃ k ∈ N k U ) = ⋃ k ∈ N A ( k U ) .

    But Y is Banach so by Baire's category theorem

    ∃ k ∈ N : ( A ( k U ) ¯ ) ∘ ≠ ∅ .

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

    B r ( c ) ⊆ ( A ( k U ) ¯ ) ∘ ⊆ A ( k U ) ¯ .

    Let v ∈ V, then

    c , c + r v ∈ B r ( c ) ⊆ A ( k U ) ¯ .

    By continuity of addition and linearity, the difference rv satisfies

    r v ∈ A ( k U ) ¯ + A ( k U ) ¯ ⊆ A ( k U ) + A ( k U ) ¯ ⊆ A ( 2 k U ) ¯ ,

    and by linearity again,

    V ⊆ A ( L U ) ¯ .

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

    ∀ y ∈ Y , ∀ ε > 0 , ∃ x ∈ X : ∥ x ∥ X ≤ L ∥ y ∥ Y and ∥ y − A x ∥ Y < ε . ( 1 )

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

    Let y ∈ V. By (1), there is some x1 with ||x1|| < L and ||y − Ax1|| < 1/2. Define a sequence {xn} inductively as follows. Assume:

    ∥ x n ∥ < L 2 n − 1 and ∥ y − A ( x 1 + x 2 + ⋯ + x n ) ∥ < 1 2 n . ( 2 )

    Then by (1) we can pick xn+1 so that:

    ∥ x n + 1 ∥ < L 2 n and ∥ y − A ( x 1 + x 2 + ⋯ + x n ) − A ( x n + 1 ) ∥ < 1 2 n + 1 ,

    so (2) is satisfied for xn+1. Let

    s n = x 1 + x 2 + ⋯ + x n .

    From the first inequality in (2), {sn} is a Cauchy sequence, and since X is complete, sn converges to some x ∈ X. By (2), the sequence Asn tends to y, and so Ax = y by continuity of A. Also,

    ∥ x ∥ = lim n → ∞ ∥ s n ∥ ≤ ∑ n = 1 ∞ ∥ x n ∥ < 2 L .

    This shows that y belongs to A(2LU), so V ⊆ A(2LU) as claimed. Thus the image A(U) of the unit ball in X contains the open ball V/2L 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

    X → X / N → α 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 (Dieudonné, 12.16.8).

    The open mapping theorem can also be stated as

    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 u ^ : X / ker ⁡ ( u ) → Y is a TVS-isomorphism onto its image.

    References

    Open mapping theorem (functional analysis) Wikipedia
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