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Power series space representations of nuclear Fréchet spaces


Author: Dietmar Vogt
Journal: Trans. Amer. Math. Soc. 319 (1990), 191-208
MSC: Primary 46A06; Secondary 46A45, 46M99
DOI: https://doi.org/10.1090/S0002-9947-1990-1008704-5
MathSciNet review: 1008704
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Abstract: Let $ E$ be a nuclear graded Fréchet space such that the norms satisfy inequalities $ \vert\vert\vert\vert _k^2 \leq {C_k}\vert\vert\vert{\vert _{k - 1}}\vert\vert\vert{\vert _{k - 1}}$ for all $ k$, let $ F$ be a graded Fréchet space such that the dual (extended real valued) norms satisfy inequalities $ \vert\vert\vert\vert _k^{*2} \leq {D_k}\vert\vert\vert\vert _{k - 1}^*\vert\vert\vert\vert _{k + 1}^*$ for all $ k$, and let $ A$ be a tame (resp. linearly tame) linear map from $ F$ to $ E$. Then there exists a tame (resp. linearly tame) factorization of $ A$ through a power series space $ \Lambda _\infty ^2(\alpha )$. In the case of a tame quotient map, $ E$ is tamely equivalent to a power series space of infinite type. This applies in particular to the range of a tame (resp. linearly tame) projection in a power series space $ \Lambda _\infty ^2(\alpha )$. In this case one does not need nuclearity. It also applies to the tame spaces in the sense of the various implicit function theorems. If they are nuclear, they are tamely equivalent to power series spaces $ {\Lambda _\infty }(\alpha )$.


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DOI: https://doi.org/10.1090/S0002-9947-1990-1008704-5
Article copyright: © Copyright 1990 American Mathematical Society

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