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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.
Let be a continuous linear map between topological vector spaces.
The continuous dual space of is denoted by
The transpose of 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 endowed with this topology is denoted by The topology is the weakest topology on making all linear functionals in continuous.
If then the polar of in is denoted by
If is a seminorm on , then will denoted the vector space endowed with the weakest TVS topology making continuous. [1] A neighborhood basis of at the origin consists of the sets as ranges over the positive reals. If is not a norm then is not Hausdorff and is a linear subspace of . If 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.
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:
Theorem [1] — If is a continuous linear map between two Fréchet spaces then the following are equivalent:
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 be a Fréchet space and be a linear subspace of The following are equivalent:
Theorem [1] — On the dual of a Fréchet space , the topology of uniform convergence on compact convex subsets of is identical to the topology of uniform convergence on compact subsets of .
Theorem [1] — Let be a linear map between Hausdorff locally convex TVSs, with also metrizable. If the map is continuous then is continuous (where and carry their original topologies).
Theorem [2] (E. Borel) — Fix a positive integer . If is an arbitrary formal power series in indeterminates with complex coefficients then there exists a function whose Taylor expansion at the origin is identical to .
That is, suppose that for every -tuple of non-negative integers we are given a complex number (with no restrictions). Then there exists a function such that for every -tuple
Theorem [3] — Let be a linear partial differential operator with coefficients in an open subset The following are equivalent:
being semiglobally solvable in means that for every relatively compact open subset of , the following condition holds:
being -convex means that for every compact subset and every integer there is a compact subset of such that for every distribution with compact support in , the following condition holds:
This article has multiple issues. Please help
improve it or discuss these issues on the
talk page. (
Learn how and when to remove these template messages)
|
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.
Let be a continuous linear map between topological vector spaces.
The continuous dual space of is denoted by
The transpose of 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 endowed with this topology is denoted by The topology is the weakest topology on making all linear functionals in continuous.
If then the polar of in is denoted by
If is a seminorm on , then will denoted the vector space endowed with the weakest TVS topology making continuous. [1] A neighborhood basis of at the origin consists of the sets as ranges over the positive reals. If is not a norm then is not Hausdorff and is a linear subspace of . If 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.
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:
Theorem [1] — If is a continuous linear map between two Fréchet spaces then the following are equivalent:
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 be a Fréchet space and be a linear subspace of The following are equivalent:
Theorem [1] — On the dual of a Fréchet space , the topology of uniform convergence on compact convex subsets of is identical to the topology of uniform convergence on compact subsets of .
Theorem [1] — Let be a linear map between Hausdorff locally convex TVSs, with also metrizable. If the map is continuous then is continuous (where and carry their original topologies).
Theorem [2] (E. Borel) — Fix a positive integer . If is an arbitrary formal power series in indeterminates with complex coefficients then there exists a function whose Taylor expansion at the origin is identical to .
That is, suppose that for every -tuple of non-negative integers we are given a complex number (with no restrictions). Then there exists a function such that for every -tuple
Theorem [3] — Let be a linear partial differential operator with coefficients in an open subset The following are equivalent:
being semiglobally solvable in means that for every relatively compact open subset of , the following condition holds:
being -convex means that for every compact subset and every integer there is a compact subset of such that for every distribution with compact support in , the following condition holds: