ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)
Lec h Dr e w n o w s k 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.
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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.
O ur 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 (V e 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
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|| : y e L } .
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].
Theorem 1. I f £ |L is X-polar, then Ç is (А л £)-polar.
P ro o f. 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 U2 e % so that
--- л
U2 c z U l and L n U 2 cz U l n L ;
this is possible because £ |L is А-polar. Next, choose U3 E Jtt such that U3 + U3 a U2. Then for every V e'V' we have
L n Ü ] cz L n { U 3 + U3 + V) cz L n ( U 2 + V) c ( L n U 2)+V.
Hence, since V was arbitrary, we get
L n Ü3 c L n U 2 cz L n U i which proves £ | L ^t|L . ■
4 — Prace Matematyczne 24.2
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As an immediate consequence of Theorem 1 we note the following Corollary 1. Let rj be a weaker linear topology on a T L S (A", f). I f there exists a dense subspace L o f (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 T V S (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.
Corollary 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 o f the same type.
P ro o f. 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. Then a ^ Я < £ | L (the strict inequality holds because x„ -» 0 in Я but not in f). To finish it suffices to set rj
= Я л £ and apply Theorem 1. ■
R e m a rk . The converse is also true: If r] as in Corollary 2 exists, then (V, £)л is non-minimal. (Use the particular case of Corollary 1.)
Since со (the space of all scalar sequences) is a unique, up to isomorphism, minimal locally convex F -space of infinite dimension (see [3], Postscript (b)), we may reformulate the locally convex version of Corollary 2 as follows:
Corollary 3. A metrizable locally convex space (X , if) admits a strictly weaker metrizable locally convex topology such that £ is rj-polar if and only if (X, f) is not isomorphic to a subspace o f со.
This condition is of course satisfied when (X, f) is non-separable or normed and dim X = oo. The result in the latter case (already obtained in Corollary 2), with y\ normed, is due to De Wilde and Tsirulnikov ([10], Proposition 2.3). Our proof for this particular case is somewhat simpler : Let || • || be a norm defining £, and let L be any infinite-dimensional separable closed subspace of (X, f). Then, using separability and the Hahn-Banach theorem, we find a sequence {f„) of linear functionals on L such that ||x|| = sup|/„(x)| for x e L . From this it follows directly that ^ | L is polar with respect to о ~ о (L, (/„)), and we finish the proof as for Corollary 2 above.
If a, Д are two linear topologies on a linear space E, then we say that /?
satisfies the filter condition with respect to a [9], if a ^ Д and every ^-Cauchy filter (or net) in E that a-converges to 0 also ^-converges to 0. We shall write a -< f when such a situation occurs. It is known (cf. [9]) that if a < /I and a is Hausdorff, then a -< if and only if the unique continuous extension j : (E , f ) A
->(£, а )л of the identity map i : (E , Д) ->(£, a) is an injection. This in turn is equivalent to the requirement that the Д-extension of a to Êp is Hausdorff.
Theorem 2. Suppose L is a subspace o f a TVS (V, if) and X is a vector topology on L such that
(*) X < t \ L .
Then
P ro o f. Let M be the closure of L in ( Xit f), and let p be the ( | | M)- extension of Я to M, i.e.,
p — X л (f\ M).
From (*) it follows that p is Hausdorff. Moreover,
( + ) P л l = X л f.
In fact, using (C) we have
p Л I = (X A (f I M)) Л f = Я A ((! I M) Л i) = A Л l Similarly, using (B), we get
(X a f ) \ X = X л
Hence, since Я a ^ f , we see that X л f is the f-extension of Я л £ to AT{.
Since p is Hausdorff, it follows from ( + ) and (A) that also Я л <f is Hausdorff, and so Я л Ç -< Ç. я
Theorem 3. Let L be a closed subspace o f an F-space (X , f) and X a metrizable vector topology on L weaker than £| L. Define ц = X л Then : (i) I f £ \L has a base at 0 consisting o f X-complete sets, then Ç has a base at 0 consisting o f ц-complete sets.
(ii) I f every Ç-bounded X-Cauchy sequence in L is X-convergent, then every Un
bounded q-Cauchy sequence in X is ц-convergent. (In other words, if Ял is X- complete for every bounded subset A in (L , £ | L), then Bn is ц-complete for every bounded subset В in (X , £).)
P ro o f, (i) Let U be a closed neighbourhood of 0 in (X, if). Choose a £- neighbourhood Ux of 0 so that
U 1 + U1 c U and и г п Ь is Я-complete.
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Let F be a balanced ^-neighbourhood of 0 such that F + F + F c t / j .
We are going to show that Vn is ^-complete and Vv cz U.
Let Q : X -> X / L be the quotient map, and note that X/ L is an F -space under the topology £/L = r\/L. Let (x„) с V be an ^-Cauchy sequence. Then (1Qx„) is a Cauchy sequence in X/ L, hence convergent to a point belonging to Q (V) cz Q (V+ W), where W is an arbitrarily chosen balanced ^-neighbourhood of 0. We shall yalso assume that W+ W cz V. Therefore, we may find z = z w e V+ W such that Q(x„ — z) -* 0. It follows that there exists a sequence (yn) in L such that
( + ) ( x „ - z ) - y H-+ 0 (£);
then rj ^ £ implies that also
( + + ) (xn- z ) - y n -+0 {rj).
For n large enough we have (x„ — z) — y„e W, whence
y „ e (x „ -z )+ W c V + V + W + W cz v+ V + V cz Ux.
We may assume that (y„) cz L n U 1; recall that L n U t is Я-complete. Now from (-{-+) and the fact that {x„ — z) is rç-Cauchy it follows that the sequence (y„) is Cauchy with respect to r]\L = À. Hence there exists y e L n U 1 such that
У п ~ * У f a ) -
Now, denoting x — y + z, we have
x H = ({xn- z ) - y n) + z-+yH -> xfa) and
x = y + z e L r \ U l + V-\-W cz U i + U ± + W, whence
xeU^ + U * c W = U.
Thus we have shown that every rç-Cauchy sequence (x„) cz V is ^/-convergent to a point x e U . Hence Vn is ^/-complete and contained in U, and this of course concludes the proof of (i).
(ii) Let a sequence (хи) с X be £-bounded and rç-Cauchy. Then as in the above proof of (i) we find z e X and a sequence (y„) in L such that ( -f ) and ( + + ) hold. The first relation implies (y„) is £-bounded, while the second one yields (y„) is t]-Cauchy. Hence the sequence (y„) is Я-convergent to a point y in L and so, finally, -► x = y + z (r7). ■
An inspection of part (i) of the above proof gives the following
Co r o l l a r y 4. Suppose the assumptions of Theorem 3 are satisfied and let V be a closed absolutely convex neighbourhood of 0 in (X , £) such that V n L is k- complete. Then Vv is q-complete and Vri <= 3F.
P r o o f is a slight modification of the argument used in proving part (i) of Theorem 3 : Omitting the selection of U and U1, take W = rV( r > 0). Then we get y„ e2(l +r) V for large n. Hence it follows that y e 2 { \ + r ) V and next x e 3 ( l + r)V. Since x does not depend on r and F is £-closed, x e 3 F . ■
We conclude the paper by considering a special case of the situation we dealt with above. Below, (X, || • ||) is a Banach space whose norm topology is ç, В is its closed unit ball, and L is a closed subspace.
From Corollary 4 we get directly the first part of our next result (cf. also the proof of Theorem 2 in [5]).
Th e o r e m 4. Suppose к is a metrizable vector topology on L weaker than Ç | L, and define q = к л Ç. I f B n L is к-complete, then U = Bn is q-complete and U cz 3B. Hence (X , || • ||) has an equivalent norm whose closed unit ball (viz., U) is rj-complete. I f in addition, (B — z ) n L is к-closed o f every z eX, then В itself is q- complete.
P ro o f. We have to show only the last part of the theorem, and for this it suffices to check that В is ^-closed. Let (x„) c= В and x„ -> x (q). Then there exists a sequence (y„) in L such that
У п ^ 0 (A = rj|L) and ||x „ - x - y „ || -+0.
Fix an r > 0. Then for n large enough we have IIx-y„W ^ 1 + r and hence
y „ e L n ( ( l +r ) B — x).
Since the set on the right-hand side of the last relation is Я-closed and y„ -> 0 in Я, we get 0 e ( l+ r ) B — x, i.e., x e ( l +r)B. Since this holds for every r > 0, we must have x e B . ■
Co r o l l a r y 5. Suppose g is a vector topology on X such that, denoting C
= B n L, we have (1) В is g-closed, (2) C is g-bounded, (3) C is g-complete and (4) g \C is metrizable.
Then there exists a weaker metrizable vector topology ц on (X, Ç) such that В is q-complete and q \C = g\ C. If, in addition, g is locally convex and (C , g | C) is separable, then ц can be chosen to be a normed topology.
P ro o f. An argument similar to that used in the proof of Theorem 4 in [6]
226 L. D r e w n o w s k i
proves the existence of a metrizable vector topology к on L such that к j C
= q\C. In view of (2) we have к ^ £ |L . Let z e X and choose к so that B — z а кВ. Since B — z is ^-closed by (1), (B — z ) n L is (g | L)-closed. But ( B - z ) n L cz (кВ) n L = kC, and on kC the topologies induced by q and к coincide and are complete, by (3), and so (B — z ) n L is 2-closed. Thus it remains to set r] = к л £ and apply Theorem 4. If the additional assumptions are also fulfilled, then к may be chosen to be a normed topology, by Theorem 1 in [6], and then tj is normed as well, ш
Corollary 5 applies in particular when X — Y* is a dual Banach space, q is its weak-star topology, and L = M 1, where M is a closed subspace of Y such that Y /M is separable and of infinite dimension. We then get the existence of a strictly weaker norm on X under which В is complete, and which induces the weak-star topology on B n L . This is equivalent to Proposition 2 in [8].
A similar application of Corollary 5 shows that if a Banach space X has an infinite-dimensional reflexive subspace L, then X has a strictly weaker norm under which В is complete, and which induces the weak topology on B n L . This is equivalent to Proposition 3 in [8].
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INSTITUTE OF MATHEMATICS, A. MICKIEWICZ UNIVERSITY, POZNAN, POLAND
