Weighted Frechet and LB-Spaces of Moscatelli Type

Yolanda Melendez

Abstract: The structure of the weighted Fréchet and LB-spaces of Moscatelli type appears when one combines both the structure of the Köthe sequence spaces [3] and the structure of Fréchet and LB-spaces of Moscatelli type, introduced by Moscatelli in 1980 [11] and developed by Bonet and Dierolf in [4, 5]. The theory of this new structure includes both theories.\par The main motivation for our research on these spaces are the questions which remain open in the theory of LB-spaces. The most important one is the question posed by Grothendieck [8] asking whether every regular LB-space is complete. This question is answered positively in our present frame here.\par This paper is divided into three sections. In the first section we introduce the weighted LB-spaces of Moscatelli type and study strictness, regularity and bounded retractivity. We also prove that these inductive limits are regular if and only if they are complete (under mild additional assumptions). In the second section we define the weighted Fréchet spaces of Moscatelli type and investigate when they are Montel, Schwartz and when they satisfy property $(\Omega_{\varphi})$ or property $(DN_{\varphi})$ of Vogt. In our third and last section we establish a certain duality between the weighted Fréchet and LB-spaces of Moscatelli type.

Classification (MSC2000): 46A12

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