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Extensions of Weaker Vector Topologies from a Subspace to the Whole Topological Vector Space

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Extensions of Weaker Vector Topologies from a Subspace to the Whole Topological Vector Space ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984) L ech D rew no w sk i (Poznan) Extensions of weaker vector topologies from a subspace to the whole topological vector space In the theory of topological vector spaces, one faces quite often the problem of finding a weaker vector topology, with some peculiar properties, on the given topological vector space. (Instead of topologies one may deal with norms as well.) It is sometimes easier (or more convenient) to construct first such a weaker topology on some subspace of the space, and after that to extend it to the entire space. There is a well-known method of accomplishing such an extension, simply by taking the finest vector topology on the space that is weaker than the original topology and coincides with the new topology on the subspace. (Only such extensions are considered below.) The question that arises then is of course whether the extended topology has the required properties. Let us mention a few examples, where such a procedure has been succesfully applied. In [1] it was used in proving that every infinite-dimensional Fréchet space, non-isomorphic to со, admits a strictly weaker complete locally convex topology with the same continuous dual. Here it was first shown that such a topology does exist on the Banach space c0 and on the nuclear Fréchet spaces with a continuous norm. In [10] it may be detected in the proof of Proposition 2.3 which asserts the existence of a strictly weaker norm on any infinite-dimensional normed space such that the original norm topology is polar with respect to the new norm topology, i.e., has a base at 0 consisting of sets closed in the new topology. Here the required new norm is first constructed on a countable-dimensional subspace of the given space. In [3] the procedure in question is used many times, e.g., in the proof of Theorem 3.3 which says, roughly speaking, that if two topological vector spaces X and Y have non- minimal (in a certain sense) isomorphic subspaces, then X x Y has a vector topology which is strictly weaker than the product topology but coincides on the factors X and Y with their original topologies. Finally, the content of Propositions 2 and 3 in [ 8] is that, on some Banach spaces, there exists a strictly weaker norm such that the original closed unit ball is complete under the new norm. In the proofs of these results the extension procedure is somewhat hidden but nevertheless it can be detected, see the proof of Theorem 2 in [5], and Theorems 3 and 4 below. 22 0 L. Drewnowski The main results of the present paper have their origins in the already mentioned results and proofs of De Wilde and Tsirulnikov [10], and Lotz, Peck and Porta [ 8]. Our Theorem 1 shows that polarity of one topology with respect to another is preserved by the extension procedure. Theorem 2 asserts the same for topologies related by the so-called filter condition. Theorems 3 and 4 concern metrizable (or normed) vector topologies linked by the property that one of them has a base at 0 consisting of sets which are complete in the other topology. It may be worth noticing that the results of this paper, with some obvious exceptions, remain valid in the more general context of topological abelian groups. Our terminology and notation are fairly standard, with this exception: We shall distinguish between linear topologies and vector topologies — the latter are required to be Hausdorff, while the former are not, in general. The same distinction will be made between a topological linear space (TLS) and a topological vector space (TVS). The completion of a TVS (X, Ç) will be denoted (X, £)A or (Х$, |). We assume throughout that L is a subspace of a TLS (X, Ç) and X is a linear topology on L such that A < É| L; that is, X is weaker (coarser) than the topology induced on L by Any additional assumptions on Ç, L or X will always be explicitly formulated when needed. It is well known (see, e.g., [l]-[3]) that there exists a finest linear topology rj on X satisfying the following two conditions: (1) V ^ É, (2) n \L = X. We shall call this (unique) topology rj the extension of X, and denote X л £ (as in [3]). The topology q can be also characterized as the unique linear topology on X that satisfies (1), (2) and is such that the quotient topologies on X/L corresponding to ц and £ are equal: (3) n/L = Ç/L. If ^ is a base at 0 for £ and У is a base at 0 for X, then the sets V+ U, with Ve V and Ue°U, are easily seen to form a base at 0 for rj. If L is a dense subspace of (X, £), then already the first two conditions (1) and (2) determine q uniquely, and in this case also the family of the sets (Ve i r) is a base at 0 for q. (In general, this family is a base at 0 for q | П.) It is obvious that if both £ and X are locally convex, then so is q. Similarly, assuming additionally that rj is Hausdorff, if both Ç and X are metrizable or normed, then so is, respectively, q. Moreover, if || • || is a Extensions of weaker vector topologies 221 norm defining Ç and | • | is a norm defining A, then a norm defining ц is given by the formula 111*111 = inf {|y| + ||x —y|| : yeL}. If I • I is chosen so that | • | ^ || • || | L, then ||| • ||| < || • || and | • | = ||| • ||| | L. We also observe that on X/L the quotient norms generated by || -1| and ||| -||| are always equal. Similar remarks can be made if || • || and | • | are T-norms defining £ and A. The following facts are also easy to verify. (A) If L is £-closed and both £ and A are Hausdorff, then so is А л (If L is not assumed ç-closed, then А л Ç is Hausdorff if and only if (А л £)\D is Hausdorff.) (B) If L is contained in a subspace M of X, then А л (£ I M) = (А л £) I M. (C) If L is contained in a subspace M of X and p is a linear topology on M such that A ^ p\L and p ^ £|M, then (A A p ) A £ = А Л (p A £ ) . If a and ft are two linear topologies on the same space, then ft is said to be a- polar if ft has a base at 0 consisting of a-closed sets. (In the terminology of [9], ft satisfies the closed neighbourhood condition with respect to a.) Our first result was inspired by Proposition 2.3 in [10]. T heorem 1. I f £ |L is X-polar, then Ç is (А л £)-polar. P roof. Let rj = А л Ç, and define т as the linear topology on X for which a base at 0 is obtained by taking the ^-closures of ^-neighbourhoods of 0. Then ц < г < Ç and t is the finest rç-polar topology weaker than ç. So wë have to show that t = Ç. Since t < £ and т/L = £/L, by a lemma due to W. Roelcke (see [4], Lemma 2.1) it suffices to check that t |L = £|L . Clearly i |L ^ £|L . Let ‘Ш and ÎT be bases of balanced neighbourhoods of 0 for ç and A, respectively. Let U1e (% and choose U2e% so that --------- л U2czUl and LnU2 czUl nL; this is possible because £|L is А-polar. Next, choose U3EJtt such that U3 + U3 a U2. Then for every V e 'V' we have LnÜ] cz Ln{U3 + U3 + V) cz Ln(U2 + V) c (LnU2)+V. Hence, since V was arbitrary, we get LnÜ 3 c LnU 2 cz LnUi which proves £ | L ^ t |L. ■ 4 — Prace Matematyczne 24.2 222 L. Drewnowski As an immediate consequence of Theorem 1 we note the following C orollary 1. Let rj be a weaker linear topology on a TL S (A", f). I f there exists a dense subspace L of (X , f) such that £ | L is (p J L)-polar, then £ is r\-polar. In particular, if rj is a weaker vector topology on a TVS (A , Ç) such that £ is rj- polar, then £ is (fj\ X^-polar. A TVS (X, £) is called an F-space if it is complete and metrizable. It is said to be minimal if it admits no strictly weaker vector topology, and non-minimal otherwise. C orollary 2. Let (X, f) be a metrizable TVS whose completion (X, £)л is non-minimal. Then (X, f) admits a strictly weaker metrizable vector topology rj such that Ç is rj-polar. Moreover, if £ is locally convex or normed, then t] can be chosen to be of the same type. Proof. By a result of Kalton and Shapiro ([7], Theorem 3.2), (X, £)л contains a regular basic sequence (x„). By ([3], Theorem 2.8), we may assume (y„) X. Since (y„) is regular, the sequence ( f „) of functionals biorthogonal to (x„) is equi-continuous on the closed linear span L of (x„) in (V, £). Moreover, if о is the weak topology on L determined by the/„’s, then о < £ | L and £ | L is easily seen to be cr-polar. Define a norm | • | on L by 00 M = Z Г " \ ш \ , n= 1 and let Я denote the topology of | • | on L.
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