Density conditions in Fréchet and (DF)-spaces

Wc survey our¡naln resulis on ihe density condition for Fréchet spaces and on the dual density conditions for (DF)-spaces (cf. [21and (31, 1.) as well as torne recent developrnents. At the end ofsection 1., we include a new result on the projective tensor product of two Fréchel spaces. Tatkinen’s construction of counterexamples to Grothendieck’s “probl~me des topologies” yields Fr¿chet spaces E, F with the density condition such that E ®f 15 not (even) distinguished (see (141). Wc prove now thai the negative solution ofthe “problérne des topologies” it, in fact, the only obsiruction: For two Fréchet spaces E and F with ihe density condition, E ~f has the densiíy condition as well (and hence it distinguished) whenever ihe “problémedes topologies” has a positive solution for the pair (E,fl. Distinguished locally convex (l.c.) spaces, titat is, locally convex spaces E sucit titat tite strong dual E’,, is barrelled, were introduced by J. Dieudonné and L. Schwartz. Later on, A. Grotitendieck sitowed titat a metrizable l.c. space E is distinguished ib and only ifE’,, is bornological. Let (U,,),N denote a basis of 0-neighboritoods for a metrizable 1.c. space E and define tite “inductive dual” E’, of E by E’»— md Et.. Titen E’ carnes a stronger topology titan Ek,,, and, in fact, E’, is tite bornological space associated witit E’,,. Hence E is distinguisited ib and only ib E’,, = E. Titus, tite class of tite distinguisited Frécitet spaces is related to tite naix’e idea titat tite strong dual oba Frécitet space E; i.e., tite strong dual of a space witicit can be represented as a countable (reduced) projective limit of Banacit spaces, sitould also be representable as a countable inductive limit of Banacit spaces (and titat tite duality sitould work by simply dualizing tite projective spectrum). As tite existence of non-distinguisited Frécitet spaces demonstrates (the first example of sucit a space was due to Kótite and Orotitendieck and was, in fact, a certain Kátite echelon space Xi), the naive idea does not work 1980 Mathematics Subject Clastification (1985 revision): 46A06, 46A07, 46A09, 46A20, 46A32, 46A45, 46M05, 46M40 Editorial de la Universidad Complutense. Madrid, 1989. http://dx.doi.org/10.5209/rev_REMA.1989.v2.18067 60 K. D. Bierstedt-Ji Bonet out in general. From titis point ofview, tite notion obdistinguisitedness for Frécitet spaces prevents a certain “patitology”, and indeed tite class ob tite distinguisited Frécitet spaces is tite largest one among alí classical subclasses arising in tite classification of Frécitet spaces. (But note titat by a very recent result of J. Taskinen [14], tite Frécitet space C(R)nL1(R), endowed witit tite natural intersection topology arising from tite compact-open topology on C(R) and tite norm topology of L1(R), is not distinguisited. Since C(R) n L,(R) is a natural space of analysis, it is quite doubtful from titis point of view witetiter non-distinguisitedness of a Frécitet space should really be considered as a “patitology”.) Tite so-called “density condition”arose brom tite study obS. Heinricit [10] on ultrapowers of l.c. spaees. Our work in [2]on titis condition started witit tite simple observation titat for metrizable l.c. spaces, tite density condition 15 closely related to tite notion ob distinguisitedness. “At tite end ob tite day”, it tumed out titat a metrizable l.c. space E satisfies tite density condition if and only ib tite space 4(E) of all absolutely summable sequences in E is distinguisited and titat titis itolds ib and only ifalí bounded subsets in tite strong dual E’5are metrizable. As a consequence of titis remarkable equivalence, we were able to citaracterize tite ecitelon spaces k=k(L A) (of arbitrary order p, 1 ~p.cocorp =0) witit tite density condition in terms of the Kótite matrices A. In tite case p= 1, witere tite density condition is equivalent to distinguisitedness, titis led to a citaracterization of tite distinguisited Kótite ecitelon spaces 2.. It was also possible to classi& tite distinguisited Kótite ecitelon spaces 2.1<E)= 2.(LA:E) = witit values in a Frécitet space E completely (witicit involves an interesting dicitotomy). Tite present survey article is based on an invited lecture (by tite firstnamed autitor) during tite Functional Analysis Conference at El Escorial, June 14, 1988. In Section 1., we repofl on tite main results of [2] and on some recent developments in titis connection, mainly due to D. Vogt [16] and J. TasUnen [14]; at tite end ob titis section, we also include a new result on tite initeritance of tite density condition under projective tensor products. Seclion 2. is devoted to a similar report on tite first pad of [3]. Tite starting point of our work in [3] was tite observation that for metrizable l.c. spaces E, tite “proper” definitions ob tite notions ob distinguisitedness and ob tite density condition involve tite strong dual E’~, of E ratiter titan E itselfand titat several of tite proobs in [2]proceeded by use ob duality titeory. Following tite patit paved by A. Orotitendieck’s introduction of tite class of (DF)-spaces, we titerefore definedand studied “dua/ densiiy conditions” mainly in tite ftamework of (Dfl-spaces. (It turned out titat (DF)-spaces, and not tite larger class ob (gDF)-spaces, are indeed tite proper context bor studies ob titis type.) Density condiíions in Frécheí and (DF)-spaces 61 Por a strong dual of a metrizable j.c. space, quasibarrelled and bornological are equivalent notions, but titese properties split up in tite class of (DF)spaces. Correspondingly, we itave a dualdensity condition (DDC) and a strong dua/ dens¡íy condition (SDDC) for (DF)-spaces. A (DF)-space satisfies (DDC) ib and only if tite bounded subsets of E are metrizable, and it also satisfies (DDC) (resp., (SDDC)) ib and only ifthe space /..,(E) ofalí bounded sequences witit values in E is quasibarrelled (resp., bornological). For a locally complete (DF)-space E and a Kótite ecitelon space 2.,, (DDC) allows to citaracterize tite (quasi-) barrelled spaces 142.,,E) ob continuous linear mappings from 2. to E, and to give “good” necessary and “good” sufficient conditions bor L«2.I,E) to be bornological. Tite s-tensor product (2.3’b ®,E and íts completion (2.)’,, ®., E are natural topological subspaces of t(2.,E) ib E is a complete 1.c. space. But, interestingly enougit, tite dual density conditions do not occur in the classification ob tite quasibarrelled and bornological (DF)-spaces of type (2.)’,,®,E and (2.,)’,, ®,, E, titey come in only witen one passes to tite “fulí” space Lb(2.I,E). In titis survey, we itave concentrated on a presentation of tite results of[2] and (ob tite first part) of [3].Among otiter titings, one metitod of [3] involves “reverse projective description”, and we state tite main result in titis direction <as Titeorem 2.7). Ideas of titis type are closely connected witit “weighted inductive limits” of spaces of continuous functions witicit, from tite sideof interesting applications, also were one of our main motivations for tite study of tite dual density conditions for (DF)-spaces. However, sketcites of proofs, or of tite metitods, and tite applications to weighted inductive limits (see tite last part ob [3]) are beyond tite scope of titis article. 1. HEINRICH’S DENSITY CONDITION FOR FRECHET SPACES Ultraproducts itave been used for sorne time in Banacit space titeory as a bramework bor some aspects of the so-called “local titeory” ob Banacit spaces. In itis article [10] (witicit raised a number of open questions and also introduced an interesting notion ob superreflexivity bor locally convex spaces), 5. Heinricit titen proceeded to study tite corresponding construction in tite context obgeneral locally convex (l.c.) topological vector spaces, witit applications to tite nonlinear (i.e., uniform and Lipscititz) classiflcation of sucit spaces. Heinricit investigated two ditlerent kinds of ultrapowers of a l.c. space E witit respect to an ultrafilter D on a set L tite fulí ultrapower (E),, and tite bounded ultrapower [E],,,a natural subspace ob (E),,. Botit definitions are canonical extensions of tite Banacit space concept: (E),, is tite maximal space to witicit tite continuous seminorms of E can be extended (and arises from tite nonstandard itulí in nonstandard analysis), while tite (new) construction of 62 K. D. Bierstedt-J Bonet [E],,is ratiter based on tite fact titat for a Banacit space 2<, tite ultrapower (X},, is generated by alí D-bounded subsets of X’/D. (See [10] for details.) Botit kinds of ultrapowers itave certain advantages, and titus one is interested in tite so-called “densiíy condiíion”for a Le. space E, a necessary and suffícent condition for tite density of tite bounded ultrapower [E],, in tite fulí ultrapower (E)» for an arbitrary “ ,c(E)-good countably incomplete” ultrafilter D. (Tite condition also occurs independently in Heinricit’s investigations on tite duality of ultrapowers.) Heinricit notes titat “tite density condition is of obvious importance in tite study and application of ultrapowers” and citaracterizes titis condition in terms of “standard” l.c. titeory as follows ([lO], Titeorem 1.4). 1. Definition (5. Hein¡icit). Let E denote a (Ha usdorff) lx. space, 91(E) (he systeni of alí closed abso/utely convex neighhorhoods ofO in E (or an>’ basis of O-neighboriroods in E) andB(E) tire systern ofal! closed abso/ute!>’ conveir and bou nded subseis ofE (or an>’ basis ofbounded seis in E). Tiren E satisfles the densiíy condiiion ¡ji given an>’ funcílon 2.: 91(E) —. RAtOI andan arbiirary y E 91(E), tirere always exis! afinite subseí U of91(E) and Be B(E) sucir thai (1) n 2.(U)UcB+V. U. U By taking polars in tite inclusion (1), it is easily seen titat tite le. space E satisfies tite density condition iband only ib for eacit 2.: 91(E).—*R4\ (0> and eacit VE 91(E), one can find Uc9/(E) finite and BEB(E) witit (2) r( u 2.(uyJ’)=r( u 2.(U)LP)DBn Y’, UEU UES¡ witere F(resp., 17) denotes tite absolute