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- The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, 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. (en)
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- 11023 (xsd:nonNegativeInteger)
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- If is a continuous linear map between two Fréchet spaces, then is surjective if and only if the following two conditions both hold:
is injective, and
the image of denoted by is weakly closed in . (en)
- If is a continuous linear map between two Fréchet spaces then the following are equivalent:
is surjective.
The following two conditions hold:
is injective;
the image of is weakly closed in
For every continuous seminorm on there exists a continuous seminorm on such that the following are true:
for every there exists some such that ;
for every if then
For every continuous seminorm on there exists a linear subspace of such that the following are true:
for every there exists some such that ;
for every if then
There is a non-increasing sequence of closed linear subspaces of whose intersection is equal to and such that the following are true:
for every and every positive integer , there exists some such that ;
for every continuous seminorm on there exists an integer such that any that satisfies is the limit, in the sense of the seminorm , of a sequence in elements of such that for all (en)
- 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 . (en)
- Let be a linear partial differential operator with coefficients in an open subset
The following are equivalent:
For every there exists some such that
is -convex and is semiglobally solvable. (en)
- Let be a linear map between Hausdorff locally convex TVSs, with also metrizable.
If the map is continuous then is continuous . (en)
- Fix a positive integer .
If is an arbitrary formal power series in indeterminants 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 . Then there exists a function such that for every -tuple (en)
- Let be a Fréchet space and be a linear subspace of
The following are equivalent:
is weakly closed in ;
There exists a basis of neighborhoods of the origin of such that for every is weakly closed;
The intersection of with every equicontinuous subset of is relatively closed in . (en)
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- Banach (en)
- E. Borel (en)
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| rdfs:comment
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- The theorem on the surjection of Fréchet spaces is an important theorem, due to Stefan Banach, 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. (en)
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- Surjection of Fréchet spaces (en)
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