The density condition in quotients of quasinormable Fréchet spaces, II
1999; Springer Science+Business Media; Volume: 12; Issue: 1 Linguagem: Inglês
10.5209/rev_rema.1999.v12.n1.17191
ISSN1988-2807
Autores Tópico(s)Fixed Point Theorems Analysis
ResumoIt ja proved that a Fréchet apare la quasinormable if, and only if, every quotient apare satiafies tbe density condition of Heinricb.This answers positively a conjecture of Bonet and Días.The chasa of quasinormable locally convex apares was introduced and atudied by Grothendieck in [10].Thla class has recently received much attention in the context of Fréchet apares and of Kóthe sequence apares (see [5, 6, 7, 8,9,15,16]).In particular Bonet, Díaz [7, 8] and Díaz, Fernández [9] proved that a Kóthe sequence apare of order p, 1 = p = oc or p = 0, is quasinormable if, and only if, every quotient apare satiafies the denaity condition of Heinrich [11].Thia result ¡a relata! to important previous theorema by Bellenot [3] and Valdivia [17], namely, a Fréchet apare ja Schwartz (respectively, totally refiexive) if every quotient ia Montel (respectively, reflexive).Accordingly, the following question was asked as Open Problem 15 in [2]: If every quotiení of o Fréchet apoce E Ita8 tIte de»s¿ty couid¿t¿on, ¿a E quasiviormable?In [1] we ahowed that the ayawer ja positive under the assumption that E ja separable.In thia note we remove thia hypotheaia hence we provide a complete solution to the just mentioned problem.In what followa we recalí sorne notat¡on.In the sequel, given a Fréchet apare E we denote by (II IIk)k an increasing fundamental ayatem of seminorma defining the topology of E auch that the seta
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