On Semi-Barrelled Spaces

ON SEMI-BARRELLED SPACES arXiv:1304.0360v5 [math.GN] 13 May 2013 HENRI BOURLÈS Abstract. The aim of this paper is to clarify the properties of semibarrelled spaces (also called countably quasi-barrelled spaces in the literature). These spaces were studied by several authors, in particular in the classical book of N. Bourbaki ”Espaces vectoriels topologiques”. However, six incorrect statements can be found in this reference. In particular: a Hausdorff and quasi-complete semi-barrelled space is complete, a semi-barrelled, semi-reflexive space is complete, a locally convex hull of semi-barrelled semi-reflexive spaces is semi-reflexive, a locally convex hull of semi-barrelled reflexive spaces is reflexive. We show through counterexamples that these statements are false. To conclude, we show how these false claims can be corrected and we collect some properties of semi-barrelled spaces. 1. Introduction Semi-barrelled spaces are studied in the classical book of N. Bourbaki [1] (”Espaces vectoriels topologiques”), in §IV.3, n◦ 1 (”Espaces semitonnelés”). A locally convex space E is said to be semi-barrelled if the following condition holds: (a) let U be a bornivorous part of E which is the intersection of a sequence of convex balanced closed neighborhoods of 0 in E; then U is a neighborhood of 0 in E. Chronologically, (DF) spaces were introduced before semi-barrelled spaces, by Grothendieck [3]: a locally convex space E is a (DF) space if (a) holds and (b) the canonical bornology of E has a countable base. Then Husain [5] considered spaces satisfying (a) but not (b) and called them ”countably semi-barrelled spaces”. Thus, these spaces are called ”semi-barrelled spaces” by Bourbaki [1], and we adopt the latter terminology in the sequel. Bourbaki’s account on semi-barrelled spaces consists of the above-quoted section, where three equivalent definitions are given, and of nine statements (Exerc. 9, p. IV.60). Six of them are not correct, therefore a clarification of the properties of these spaces is needed and presented below. These statements are the following ones: 2010 Mathematics Subject Classification. Primary 57N17; Secondary 54A10. Key words and phrases. semi-barrelled space, countably quasi-barrelled space, espace semi-tonnelé. 1 2 H. BOURLÈS (1) Let E be a locally convex Hausdorff semi-barrelled space, M be a closed subspace of E, E ′ be the dual of E. The strong topology β (M 0 , E/M) is identical to the topology induced on M 0 by the strong topology β (E ′ , E) . (2) Let E be a locally convex Hausdorff space, M a (non necessarily closed) vector subspace of E. If M is a semi-barrelled space, then the strong topology β (E ′ /M 0 , M) is identical to the quotient topology by M 0 of the strong topology β (E ′ , E) . (3) A Hausdorff and quasi-complete semi-barrelled space M is complete. (4) Let E be a semi-barrelled Hausdorff space and M be a closed subspace of E. Then E/M is a semi-barrelled space. (5) Let (En ) be a sequence of semi-barrelled spaces, E be a vector space, and for each n, let fn be a linear mapping from En into E. Suppose that E is the union of the fn (En ). Then E is semi-barrelled for the finest locally convex topology for which all the fn are continuous. (6) The completion of a semi-barrelled Hausdorff space is semi-barrelled. (7) A semi-barrelled, semi-reflexive space M is complete. (8) Let (En ) and E be as in (5). If each En is semi-reflexive and if E is Hausdorff, then E is semi-reflexive. (9) Let (En ) and E be as in (5). If each En is reflexive and if E is Hausdorff, then E is reflexive. Claims (4) and (5) were proved in ([5], Thm. 8 and Corol. 14) with the sequences (En ) and (fn ) replaced by non-necessarily countable families (Eα ) and (fα ) in Claim (5) –and one can notice that it is not necessary to assume in Claim (4) that M is closed. Claim (6) was proved in ([6], Chap. V, Prop. 3, p. 133). We show below through counterexamples that Claims (1)-(3) and (7)-(9) are false. So, Bourbaki’s account on semi-barrelled spaces is very misleading for people studying topological vector spaces, and a clarification of the properties of these spaces is needed. We conclude by additional remarks where we show how these false claims can be corrected and we collect some properties of semi-barrelled spaces. 2. Counterexamples 2.1. Counterexample to Claim (1). An example is given in ([11], §27, 2., p. 370) of a Montel space F and a closed subspace H of F , such that the initial topology of F does not induce on H its strong topology. So, let E be the strong dual of F and M = H 0 . Then E is a Montel space, thus it is barrelled, M is a closed subspace of E, H = M 0 according to the ON SEMI-BARRELLED SPACES 3 bipolar theorem, and the strong topology β (M 0 , E/M) is not identical to the topology induced on M 0 by the strong topology β (E ′ , E) . 2.2. Counterexample to Claim (2). It is well-known that there exist non-distinguished Fréchet spaces ([11], §31, 7., p. 435). Such a space M can be embedded in the product E of a family of Banach spaces ([11], §18, 3., (7), p. 208). Then the strong dual Eβ′ is a locally convex direct sum of Banach spaces, thus is barrelled; therefore Eβ′ /M 0 (the topology T1 of which is the quotient topology of β (E ′ , E) by M 0 ) is barrelled. Since M is nondistinguished, its strong dual (the topology T2 of which is β (E ′ /M 0 , M)) is not barrelled. Therefore the topologies T1 and T2 are not identical. 2.3. Counterexample to Claims (3) and (7). An example of a noncomplete Montel space has been given in ([10], §5). This space is barrelled and reflexive, hence quasi-complete ([1], p. IV.16, Remarque 2). Therefore, there exist reflexive (thus quasi-complete) barrelled spaces which are not complete. 2.4. Counterexample to Claims (8) and (9). If Claim (8) is correct, so is Claim (9) since E is barrelled if the spaces En are barrelled ([11], §27, 1., (3), p. 368). However, in ([11], §31, 5., p. 434), an example is given of a Fréchet-Montel space E1 = λ and a closed subspace N of E1 such that E = E1 /N is topologically isomorphic to l1 , thus is not reflexive. The space E1 is reflexive and E = f1 (E1 ) , where f1 is the canonical surjection, is not. Therefore, Claim (9) is false and Claim (8) is false too. 3. Concluding remarks According to the proof of ([3], corol., p. 79), Claims (8) and (9) are correct if E and the sequence (En ) are as in (5), assuming that every bounded part of E is included in the closed balanced convex hull of a finite number of fn (Bn ) where each Bn is a bounded part of En (in particular, this holds if E is a regular inductive limit of the En [2], and a fortiori if the En are closed subspaces of E and E is the strict inductive limit of the En ). On the other hand, all statements are correct if (DF) spaces are considered in place of semi-barrelled spaces ([3], prop. 5, p. 76 and corol., p. 79), ([4], corol. 2, p. 170). A closed subspace F of a semi-barrelled space E is not necessarily semibarrelled, as shown in ([3], p. 97) and ([7], (iii)), except if F is finitecodimensional ([15], Thm. 6, p. 169) or if F is countable-codimensional and such that for every bounded subset B of E, F is finite-codimensional 4 H. BOURLÈS in the space spanned by F ∪ B ([14], Thm. 3). Every infra-barrelled (also called quasi-barrelled ) space is semi-barrelled ([5], Prop. 2, p. 292), every infra-barrelled space is a Mackey space ([4], p. 107), but a semi-barrelled space is not necessarily a Mackey space by ([3], Remarque 8, p. 74) – since neither is a (DF) space. Further results on semi-barrelled spaces can be found in, e.g., [13], [12], [8], and ([9], Chap. III, p. 33 and n◦ 41., 44.; Chap. IV, 16.; Chap. VI, 4.). References [1] N. Bourbaki, Espaces vectoriels topologiques, Springer (2007). [2] K. Floret, ”On Bounded Sets in Inductive Limits of Normed Spaces”, Proc. Amer. Math. Soc., 75(2), 221-225 (1979). [3] A. Grothendieck, ”Sur les espaces (F ) et (DF)”, Summa Brasiliensis Mathematicae, 3(6), 57-123 (1954). [4] A. Grothendieck, Topological Vector Spaces, Gordon and Breach (1973). [5] T. Husain, ”Two New Classes of Locally Convex Spaces”, Math. Annalen, 166, 289-299 (1966). [6] T. Husain, S. M. Khaleelulla, Barrelledness in Topological and Ordered Vector Spaces, Springer Verlag (1978). [7] S. O. Iyahen, ”Some remarks on countably barrelled and countably quasibarrelled spaces”, Proc. Edinburgh Math. Soc., 15, 295-296 (1966). [8] A. K. Katsaras, V. Benekas, ”Sequential Convergence in Topological Vector Spaces”, Georgian Mathematical Journal, 2(2), 151-164 (1995). [9] S. M. Khaleelulla, Counterexamples in Topological Vector Spaces, Springer-Verlag (1982). [10] Y. Komura, ”Some examples on Linear Topological Spaces”, Math. Annalen, 153, 150-162 (1964). [11] G. Köthe, Topological Vector Spaces I, Springer-Verlag (1969). [12] D. Krassowska; W. Śliwa, ”When (E, σ(E, E ′ )) is a DF -space?” Commentationes Mathematicae Universitatis Carolinae, 33(1), 43–44 (1992). [13] S. Radenović, ”Some remarks on the weak topology of locally convex spaces”, Publications de l’Institut mathématique, 44(58) , 155-157 (1988). [14] J. H. Webb, ”Countable-Codimensional Subspaces of Locally Convex Spaces”, Proc. Edinburgh Math. Soc.,Series 2, 18(3), 167-172 (1973). ON SEMI-BARRELLED SPACES 5 [15] J. H. Webb, ”Finite-Codimensional Subspaces of Countably QuasiBarrelled Spaces”, J. London Math. Soc., 2(8), 630-632 (1974). Satie, ENS de Cachan/CNAM, 61 Avenue Président Wilson, F-94230 Cachan, France E-mail address: henri.bourles@satie.ens-cachan.fr