Surjection of Fréchet spaces

The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach,[1] that characterizes when a continuous linear operator between Fréchet spaces is surjective.

The importance of this theorem is related to the open mapping theorem, which states that a continuous linear surjection between Fréchet spaces is an open map. Often in practice, one knows that they have a continuous linear map between Fréchet spaces and wishes to show that it is surjective in order to use the open mapping theorem to deduce that it is also an open mapping. This theorem may help reach that goal.

Preliminaries, definitions, and notation

Let be a continuous linear map between topological vector spaces.

The continuous dual space of is denoted by

The transpose of L is the map defined by If is surjective then will be injective, but the converse is not true in general.

The weak topology on (resp. ) is denoted by (resp. ). The set X endowed with this topology is denoted by The topology is the weakest topology on X making all linear functionals in continuous.

If then the polar of S in Y is denoted by

If is a seminorm on X, then will denoted the vector space X endowed with the weakest TVS topology making p continuous.[1] A neighborhood basis of at the origin consists of the sets as r ranges over the positive reals. If p is not a norm then is not Hausdorff and is a linear subspace of X. If p is continuous then the identity map is continuous so we may identify the continuous dual space of as a subset of via the transpose of the identity map which is injective.

Surjection of Fréchet spaces

Theorem[1] (Banach)  If is a continuous linear map between two Fréchet spaces, then is surjective if and only if the following two conditions both hold:

  1. is injective, and
  2. the image of denoted by is weakly closed in (i.e. closed when is endowed with the weak-* topology).

Extensions of the theorem

Theorem[1]  If is a continuous linear map between two Fréchet spaces then the following are equivalent:

  1. is surjective.
  2. The following two conditions hold:
    1. is injective;
    2. the image of is weakly closed in
  3. For every continuous seminorm p on X there exists a continuous seminorm q on Y such that the following are true:
    1. for every there exists some such that ;
    2. for every if then
  4. For every continuous seminorm p on X there exists a linear subspace N of Y such that the following are true:
    1. for every there exists some such that ;
    2. for every if then
  5. There is a non-increasing sequence of closed linear subspaces of Y whose intersection is equal to and such that the following are true:
    1. For every and every positive integer k, there exists some such that ;
    2. For every continuous seminorm p on X there exists an integer k such that any that satisfies is the limit, in the sense of the seminorm p, of a sequence in elements of X such that for all i.

Lemmas

The following lemmas are used to prove the theorems on the surjectivity of Fréchet spaces. They are useful even on their own.

Theorem[1]  Let X be a Fréchet space and Z be a linear subspace of The following are equivalent:

  1. Z is weakly closed in ;
  2. There exists a basis of neighborhoods of the origin of X such that for every is weakly closed;
  3. The intersection of Z with every equicontinuous subset E of is relatively closed in E (where is given the weak topology induced by X and E is given the subspace topology induced by ).

Theorem[1]  On the dual of a Fréchet space X, the topology of uniform convergence on compact convex subsets of X is identical to the topology of uniform convergence on compact subsets of X.

Theorem[1]  Let be a linear map between Hausdorff locally convex TVSs, with X also metrizable. If the map is continuous then is continuous (where X and Y carry their original topologies).

Applications

Borel's theorem on power series expansions

Theorem[2] (E. Borel)  Fix a positive integer n. If P is an arbitrary formal power series in n indeterminants with complex coefficients then there exists a function whose Taylor expansion at the origin is identical to P.

That is, suppose that for every n-tuple of non-negative integers we are given a complex number (with no restrictions). Then there exists a function such that for every n-tuple p of non-negative integers.

Linear partial differential operators

Theorem[3]  Let D be a linear partial differential operator with coefficients in an open subset The following are equivalent:

  1. For every there exists some such that
  2. U is D-convex and D is semiglobally solvable.

D being semiglobally solvable in U means that for every relatively compact open subset V of U, the following condition holds:

to every there is some such that in V.

U being D-convex means that for every compact subsets and every integer there is a compact subset of U such that for every distribution d with compact support in U, the following condition holds:

if is of order and if then

See also

References

    1. Trèves 2006, pp. 378-384.
    2. Trèves 2006, p. 390.
    3. Trèves 2006, p. 392.

    Bibliography

    • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
    • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
    • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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