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.

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 |