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Stojan Radenovi C Zoran Kadelburg Scientiae Mathematicae Vol.1, No. 11998, 43{49 43 ON A. GROTHENDIECK'S PROBLEMS CONCERNING F AND DF{SPACES Stojan Radenovic Zoran Kadelburg For non-exp erts only Received Novemb er 7, 1996; revised March 6, 1997 Abstract. This is an exp ository article devoted to the answers to the ten op en problems from the fundamental pap er [16] of A. Grothendieck, some of whichwere given in the recent years. A signi cant contribution to the development of the theory of lo cally convex spaces lcs , as a part of functional analysis, was given in the fties by a group of French mathematicians L. Schwartz, N. Bourbaki, J. Dieudonn e, A. Grothendieck. The dominant part in their work, concerning this part of functional analysis, was devoted to the so called duality theory.Itwas esp ecially applied to the investigation of metrizable, i.e. Fr echet lcs, which are \nearest" to the normed and Banach spaces [14]. The article [16] of A. Grothendieck \on F and DF spaces" takes a sp ecial place in the theory of lcs. Apart from solving nearly all the problems of L. Schwartz and J. Dieudonn e, p osed in [14], a signi cant class of spaces|\DF-spaces"|was intro duced in it. This class contains a lot of imp ortant functional spaces|we mention just a few, following the pap er [16] itself: m 0 0 m 0 a The spaces E , E , D , D , intro duced byL.Schwartz in [28] for ap- L L p p plication in the theory of distributions; also the space C of continuous functions with compact supp ort. b The spaces of lo cally-analytic functions H K . c A lot of examples of functional spaces de ned by the condition that their elements b elong to at least one normed space from a certain sequence. d Miscellaneous examples of the spaces of linear mappings LE; H , where E is an F-space, and H is a DF-space, with the top ology of b ounded convergence. Investigating the features of F-spaces and the new class of DF-spaces, a lot of new problems arose, some of which remained unsolved. These unsolved questions there are 10 of them are given at the end of the pap er [16]. These questions, together with a lot of others whichwere p osed later in connection with F- and DF-spaces and their generalizations, played in the past 40 years one of the main themes to b e considered by many mathematicians in this eld. The present article is of exp ository character and its goal is to show the actuality of Grothendieck's work. Namely, some of the mentioned ten problems were solved only in the recentyears. 1991 Mathematics Subject Classi cation. 46A04, 46A03. Key words and phrases. F-spaces, DF-spaces. This researchwas supp orted by Science Fund. of Serbia, grantnumb ers 04M01 and 04M03 . 44 STOJAN RADENOVIC, ZORAN KADELBURG We b egin with a brief survey of the basic prop erties of a metrizable lcs, its strong dual and bidual. So the need for intro ducing the class of DF-spaces will b e clear. If E; t is a metrizable lcs, then it has a fundamental sequence of b ounded subsets if and only if it is normed. If B is a sequence of b ounded sets in a metrizable lcs, then there n S 1 exist scalars > 0 so that B is a b ounded subset in the space E . The strong dual n n n n=1 of each metrizable lcs has a fundamental sequence of b ounded subsets and it is metrizable itself if and only if the given space is normed. If E; t is a metrizable lcs, then the so called natural top ology in its bidual the top ological dual of its strong dual space is equal to the strong top ology.For each metrizable lcs the following two equivalent conditions of its strong dual are the main motivation for intro ducing the class of DF-spaces: S 1 0 1 If a union A = A of countably many E ;E-equicontinuous subsets of the n n=1 00 00 0 0 00 space E is E ;E -b ounded, then A is itself a E ;E-equicontinuous subset of E . 0 2 If V is a sequence of closed absolutely convex E ;E-neighb ourho o ds of origin, n T 1 0 such that the intersection V = V is an absorb ent subset in the top ological dual E , n n=1 0 then V is also a E ;E-neighb ourho o d of origin. 1. One of the direct corollaries of the prop erty 1 or the equivalent prop erty 2 is that ^ each separable b ounded subset of the completion E of the metrizable lcs E , is contained in the closure of some b ounded set from E . The problem 1 in the Grothendieck's pap er now reads as follows: ^ Is each bounded subset of the completion E of the metrizable lcs E contained in the closure of some bounded subset of the space E i.e. can the separability condition bedropped in the previous proposition? The rst negative answer to this question was given by J. Dieudonn e [13], using the continuum-hyp othesis. J. Amemiya did that without continuum-hyp othesis in the pap er [3]. See also G. Kothe [21], p. 404. 00 0 0 0 ^ 2. When metrizable lcs 's are concerned, we always have E E =E ; E ;E . In the 00 0 ^ general case, E and E are uncomparable subspaces of the algebraic dual E . Therefore 0 0 it is natural to consider spaces E; t for whichE; E ;E is a barrelled space, i.e. for 00 0 00 which each E ;E -b ounded subset of the bidual E is contained in the closure of some 00 0 t-b ounded subset the closure is taken with resp ect to the top ology E ;E . Such spaces are called distinguished. Grothendieck showed the following: 0 0 3 For each metrizable lcs E , the space H =E ; E ;E is barrelled i.e. E is distin- guished if and only if H is b ornological and if and only if H is quasibarrelled. In connection with this characterization of the strong dual of an F-space, the problem 2was formulated: Is for each F-space E the strong topology of its dual equal to the Mackey topol- 0 0 00 ogy, i.e. is E ;E=E ;E ? Of course, in the case when the F-space E is semi-re exive, the answer to this question is p ositive. But, the negative answer in the general case was given byY.Komura in the pap er [20]; see also G. Kothe [21], pp. 388 and 404. 3. The condition 1 inspired Grothendieck to de ne the class of DF-spaces. Thus, an lcs E; t is a space of typ e DF if it has a fundamental sequence of b ounded subsets and if each strongly b ounded union of countably many t-equicontinuous subsets of its dual is t-equicontinuous. Of course, each normed space is of that kind. So is the strong dual of a ON A. GROTHENDIECK'S PROBLEMS 45 metrizable lcs. It can b e easily proved that each bidual with strong top ology of a DF- space is also a DF-space; S. Dierolf [10] proved that the same is true when the bidual is endowed with the natural top ology. One of the imp ortant prop erties of strong duals of metrizable spaces whichwere proved by Grothendieck is the ab ove mentioned prop erty 3. The question 3 read as follows: Is the mentionedproperty 3 valid for an arbitrary DF-space H ? Particularly, is every quasibarrel ledspace H bornological? The negative answer to this question was given in the same year when Grothendieck's pap er was published. Examples of barrelled even complete spaces which are not b ornological were indep endently constructed byL.Nachbin [25] and T. Shirota [29]. Later Y. Komura [19] constructed an example of a barrelled DF-space which is not b ornological. Unfortu- nately Komura's pro of contained a gap. Examples of quasibarrelled and even barrelled DF-spaces which are not b ornological were given byM.Valdivia [33], [35]. 4. In the theorem 5 of his pap er, Grothendieck proved that each DF-space whose b ounded subsets are metrizable is quasibarrelled. A criterion for an F-space to b e distinguished can b e easily derived from there|it is sucient that the b ounded subsets of its strong dual are metrizable. It was natural to p ose the question of inverse statements, i.e.: 0 Arebounded subsets of the strong dual E metrizable for each distinguishedspace E of the type F? More general ly, if H is a quasibarrel led DF-space, areall of its bounded subsets metrizable? The negative answer to the p osed question was given by I. Amemiya in the ab ove mentioned pap er [3]; see also G. Kothe [21], p. 405. We mention that K.-D. Bierstedt and J. Bonet in [4] showed that for a metrizable lcs E the prop erty that b ounded sets in its strong dual are metrizable is equivalenttothe imp ortant prop erty that E satis es Heinrich's density condition. 5. If the answer to the question no. 4 had b een p ositive, the next question would not have b een p osed|it would have the p ositive answer.
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