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- characterization of surjectivity (en)
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- 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)
- 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)
- 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 . 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)
- 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)
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- Banach (en)
- E. Borel (en)
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- Surjection of Fréchet spaces (en)
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