The derived functors of Hom in the category of locally convex spaces
by
Jochen Wengenroth
University Trier

The derived functors Ext^k(E, ·) of the assignment F --> L(E, F), the space of continuous linear operators from E to F, have been introduced by Palamodov using injective resolution. Their importance is due to the fact that Ext¹(E, F)=0 holds if and only if each exact sequence of the form 0 --> F --> G --> E --> 0 splits.

After reviewing some results of Palamodov, Vogt, and Frerick and the speaker we concentrate on a conjecture of Palamodov: is it true that Ext^k(E, F)=0 holds for all k whenever E is a Fréchet space, F is a complete (LB)-space, and one of them is nuclear?

We present a positive result (obtained jointly with L. Frerick) that Ext¹(\omega, F)=0 for all complete (LB)-spaces satisfying the dual density condition of Bierstedt and Bonet (where \omega is the Fréchet space of all scalar sequences), and we then use this result to obtain a negative one: assuming the continuum hypothesis, we have Ext²(\omega, j) =/= 0, where j is the strong dual of \omega.

Date received: March 29, 2001