We exhibit examples of Fréchet Montel spaces $E$ which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2, they are of independent interest and show for example that the canonical inclusion between James spaces $J_p \subset J_q, 1 < p < q < \infty,$ is strictly cosingular. This result requires a careful analysis of the block basic sequences of the canonical basis of the dual $J'_p$ of the James space $J_p$, and permits us to show that the Fréchet space $J_{p^+} = \cap_{q>p}J_q$ has no infinite-dimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no infinite-dimensional Banach subspaces.
Fréchet spaces with no infinite-dimensional Banach quotients
ALBANESE, Angela Anna;
2012-01-01
Abstract
We exhibit examples of Fréchet Montel spaces $E$ which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2, they are of independent interest and show for example that the canonical inclusion between James spaces $J_p \subset J_q, 1 < p < q < \infty,$ is strictly cosingular. This result requires a careful analysis of the block basic sequences of the canonical basis of the dual $J'_p$ of the James space $J_p$, and permits us to show that the Fréchet space $J_{p^+} = \cap_{q>p}J_q$ has no infinite-dimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no infinite-dimensional Banach subspaces.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
