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Power series space representations of nuclear Fréchet spaces. (English) Zbl 0724.46007
The main result of this very interesting paper is a general theorem, that under certain conditions a continuous linear map A from a Fréchet space F to a Fréchet space E factors through a power series space of infinite type, and the characteristic of continuity can be estimated. The conditions are satisfied, for instance, if A is a tame linear map from an ($$\Omega$$)-space F to a nuclear (DN)-space E. If in particular A is a tame quotient map, this means that E is tamely equivalent to a power series space of infinite type. This applies in particular to the ranges of tame projections in power series spaces $$\Lambda^ 2_{\infty}(\alpha)$$. The situation in finite type power series spaces is much easier [see B. S. Mityagin and G. M. Henkin, Russian Math. Surveys 26, 99-164 (1971; Zbl 0245.46027)]. The above result for infinite type spaces is a very important step forward. It is also shown that certain nuclear spaces admitting a family of smoothing operators in the sense of J. Moser [Proc. Nat. Acad. Sci. USA 47, 1824-1831 (1961; Zbl 0104.305)] are tamely equivalent to power series spaces $$\Lambda^ 2_{\infty}(\alpha)$$.
Reviewer: F.Haslinger (Wien)

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 58C15 Implicit function theorems; global Newton methods on manifolds 46A04 Locally convex Fréchet spaces and (DF)-spaces
##### Citations:
Zbl 0245.46027; Zbl 0104.305
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