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Some results on continuous linear maps between Fréchet spaces. (English) Zbl 0585.46060
Functional analysis: surveys and recent results III, Proc. Conf., Paderborn/Ger. 1983, North-Holland Math. Stud. 90, 349-381 (1984).
[For the entire collection see Zbl 0539.00012.]
The paper is concerned with the derived functors $$Ext^ 1(E,.)$$ of the functor L(E,.), E fixed, acting from the catogory of Fréchet spaces to the category of linear spaces with the condition that all exact sequences $0\to F\to G\to E\to 0$ split and with the applications of this functors.
In the first section the short introduction to the theory of the functors $$Ext^ 1(E,.)$$ with special emphasis on the concrete representations for $$Ext^ 1(E,F)$$ and the connecting maps in case E or F is nuclear. In the third section the extension of the Grothendieck result concerning elliptic operators, that a hypoelliptic partial differential operator on $${\mathbb{R}}^ n$$ with constant coefficients has no right inverse in $$C^{\infty}(\Omega)$$, $$\Omega \subset {\mathbb{R}}^ n$$ open. In the last two sections is shown that the knowledge of condition $$Ext^ 1(E,F)=0$$ can be used for the investigation of topological properties of spaces $$E_ b\otimes_{\pi}F=L_ b(E,F)$$, E,F Fréchet spaces, one of them nuclear and the derived results are used for the investigation of the main problem.
Reviewer: J.Vaníček

##### MSC:
 46M15 Categories, functors in functional analysis 46M05 Tensor products in functional analysis 46A04 Locally convex Fréchet spaces and (DF)-spaces 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 65H10 Numerical computation of solutions to systems of equations