Fréchet spaces of Moscatelly type

Abstract

A certain class of Frechet tpaces, called of Moscatelli type, it introduced and studied. tising sorne shifiing device these Fr?chet spaces are defined as projective limits of Hanach spaces L«XDSE N), where L isa normal Banach tequence space and the X~’s are Banach tpaces. The duality between Frechet and (LB)-spacet of Moscatelli type it eslablished and the following properties of Frechet tpacet are characterized in the present context: distinguishedness, quasinonnability, Heinrich’s density condition, existence of a continuous noran in the space Gr the bidual, and the properties (DN) and (O) of Vogt. Tite aim of titis anide is to study a class of Frecitet spaces witicit itas been used recently quite obten to find counterexamples titat solved several open questions in tite titeory ob Frecitet and (DF)-spaces. In 1980, Moscatelli [13] introduced a certain type of Frecitet and (LB)spaces to find a twisted quojection, i.e., a Frecitet space witicit is a surjective limit of Banacit spaces, wititout a continuous norm but not isomorpitic to a product of Frecitet spaces eacit itaving a continuous norm. Sucit a space cannot itave an unconditional basis, according to [7]. Tite natural extension of tite classieal idea of Moscatelli yields tite following construction. We start with a normal Banach sequence space (L, II II), two sequences of Banach spaces (XJ5,,. (flPc.N and linear continuous maps f, Y5 —*2 ’, and 80 1 Bonet-S. Dierolf foreveryice Rl lei f& Yk—*Xk be a /inearmapsucir thatf,(Bk)cAk. Foreveryn e Rl, tire space F.,:=L((Yk,sk), .,) is a Banacir space according lo 1.2, and tire linear inap g,:F.,~1—*F.,, (zjk,N—*((zD,.C,,, f.,(z,,), (zk)k>,) is non decreasing. Tire Frechel space ofMoscaie//i iype associated lo (or wfth respect lo-w. rl.) (L,ii II), (Xkrjk,N, ~ andf,Y,.,—>X?icE Rl) is tire Freciret space dejinedby F= proj ((F9,,,~,(g,,).,~~) i.e., tire projective /imit of tire projective sequence of fianacir spaces (F.,).,,~ with /inking niaps (g,,),,~. 14. Proposition. Tire Frecirel space ofMoscatelil iype defined aboye coincides a/gebraically with {y=(yD,~N eflY, ’ w.ni. tire inc/usion j:F.—*f1(Y~,s~) and tire /inear niap kc N J?F—*L((Xk,rk)kEN). fty):=(f(yj)5,~. Proof. F coincides witit tite space ((zOk.Ne 1117 : gjzJ~’)=z’for al/le Rl>, enjcN dowed witit tite topology induced by [117. We denote by H tite space JEN (ye ~ : ~&k))k.~ E L((XS,rD,,N)> endowed witit tite initial topology w.r.t. tite ker’J linear mapsj and ji Define ~rF—4’Hby w((zJ)~CN).~(zk+L) . It isa direct matter to citeck titat i}i is linear, bijective and continuous (see [4,(l.3)(3)]). Since F and H are botit Frecitet spaces, w is also open tite proof is complete. a From now on we sitali make tite identificationindicated in Proposition 1.4. We close titis section recalling tite definition ob (LB)-spaccs ob Moscatelli type and some of titeir properties from [4]. Let (L,II II) be a normal Banacit sequence space. Let ,,) is a Banacit space, nc Rl. Tite corresponding(LB)-space of Moscatelil type is defined by E: = md E.,. Tite closed unit balI of E,, wifl be denoted by JS,. A basis ob 0-neigbbouritoods (0-ngitbs) in E is given by tite sets of tite bomi @s,,A,-~-8~%, 8>0, 45’ 0(ke Rl). N If(L,li II) satisfies property (‘y), titen E is regular it’ and only ib E Rl ~ 1 Vic~n B,,cr pR,, witere B,, denotes tite closure of fi,, in (X,,,r,,). In [4,Section 3] we associated to E a projective limit in tite following way. Qiven 8>0 and EK>O(ice Rl), tite Minkowski bunctional of s,,A,+BB, is denotad by p~8, and it is a norm on X,< equivalent to r,,. Titen E is tite projective limit Frecher spaces ofMoscalelil type 81 Ch L(<XbPEDSFN). 5(8~) Tite (LB)-space E is continuously injected in E and E is a complete (DF)space. A basis of 0-nghbs in E is given by tite sets