On prequojections and their duals

Abstract

Tbe paper is devoted to the class of Fréchet spaces which are called prequojections. This class appeared in a natural way in tSe structure theory of Fréchet spaces. TSe structure of prequojections was studied by G. Metafune and V.B. Moscatelli, who also gaye a survey of tSe subject. Answering a qu~tion of these authors we show tbat their result on duals of prequojections cannot be generahized from the separable case to the case of spaces of arbitrary cardinality. Wc also introduce a special class of prequojections, we calI them canonical, and show that in tSe main result of G. Metafune and V.B. Moscatelli on tSe existence of a prequojection with a given dual we may require this prequojection to be a canonical one.The main purpose of tuis paper is to iravestigate a class of Fréchet spaces which are called prequojections. This class appeared naturally iii tSe development of the structure theory of Fréchet spaces.Let us recail sorne definitions. A Fréchet space F is called a quojection if there exists a sequerace {P»J~of Banach spaces aud a sequence of linear continuous surjective mappings 1?,, P».~~14, (n E IV) such that F is isomorphic to the projective limit projn (Fn,1?»). According to [6]

Weak$^*$ closures and derived sets in dual Banach spaces

“…The study of weak * derived sets was initiated by Banach and continued by many authors, see [17,2,18,34,9,21,23], and references therein. Weak * derived sets and their relations with weak * closures found applications in many areas: the structure theory of Fréchet spaces (see [1,3,5,19,20,22,24]), Borel and Baire classification of linear operators, including the theory of ill-posed problems ( [28,31,32,33]), Harmonic Analysis ( [12,16,18,29]), theory of biorthogonal systems ( [10,30]; I have to mention that the historical information on weak * sequential closures in [10] is inaccurate). The survey [25] contains a historical account and an up-to-date-in-2000 information on weak * sequential closures.…”