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Dopovidi of the Academy of Sciences of the Ukraine, Ser. A

arXiv:math/9303203v1 [math.FA] 29 Mar 1993Dopovidi of the Academy of Sciences of the Ukraine, Ser. A(1987), No.

10, p. 9-12.M.I.OSTROVSKIIW ∗-DERIVED SETS OF TRANSFINITE ORDER OF SUBSPACESOF DUAL BANACH SPACESLet X be a separable Banach space and X∗be its dual space. If Γ is a linear (notnecessarily closed) subspace of X∗, then the w∗-derived set of Γ, denoted Γ(1), is the setof all limits of w∗-convergent sequences in Γ.

Generally speaking, Γ(1) may be non-closedin w∗-topology [1]. This was a reason to S.Banach to introduce [2, p. 213] derived setsof other orders, including transfinite ones.

Their inductive definition is the following.The w∗-derived set of order α is the setΓ(α) =[β<α((Γ(β))(1).We have Γ ⊂Γ(β) ⊂Γ(α) for β < α and if Γ(α) = Γ(α+1) then all subsequent derived setsare equal to each other. In [2, p. 209–213; 3–7] it was proved that for some separableBanach spaces there exist subspaces in their duals for which the chains of differentw∗-derived sets is long in certain sense (see Commentary).

We shall also consider onlyseparable spaces.Let us recall that Banach space X is called quasireflexive if dim(X∗∗/π(X)) < ∞,where π : X →X∗∗is the canonical embedding.THEOREM. Let X be a nonquasireflexive separable Banach space.

Then for everycountable ordinal α there is a linear subspace Γ ⊂X∗for which Γ(α) ̸= Γ(α+1) = X∗.Remarks. Let X be a separable Banach space.

Then for every subspace Γ ⊂X∗thereexists a countable ordinal α for which Γ(α) = Γ(α+1) [8, p. 50]. In [6] it is proved thatthe length of the chain of different w∗-derived sets cannot be equal to a limit ordinal.From [2, p. 213] and well-known properties of quasireflexive spaces (see [8, Chapter 4])it follows that if X is a quasireflexive separable Banach space then for every subspaceΓ ⊂X∗we have Γ(1) = Γ(2).

Hence our result is the best possible for separable Banachspaces.Lemma 1. Let X be a separable Banach space, Y be its subspace, ξ : Y →X bethe operator of the identical embedding, and Γ be a subspace of Y ∗.

Then for everyordinal α we have (ξ∗)−1(Γ(α)) = ((ξ∗)−1Γ)(α).Proof. The inclusion (ξ∗)−1(Γ(α)) ⊂((ξ∗)−1Γ)(α) follows by the w∗-continuity of theoperator ξ∗.

In order to prove the inverse inclusion it is sufficient to show that forevery sequence {fi}∞i=1 ⊂Y ∗with w∗−lim fi = f and any g ∈(ξ∗)−1({f}) there ex-ist gi ∈(ξ∗)−1({fi}) such that w∗−lim gi = g. Let {ei}∞i=1 be a sequence for whichei ∈(ξ∗)−1({fi}) and ||ei|| = ||fi||. Then the sequence {ei −g}∞i=1 is a bounded se-quence of X∗and all its limit points are in (ξ∗)−1({0}).

Weak∗topology is metrizableon bounded subsets of the dual of separable Banach space. Therefore there exists asequence {hi}∞i=1 ⊂(ξ∗)−1({0}) for which w∗−lim(ei −g −hi) = 0.

It is clear that thevectors gi = ei −hi(i ∈N) forms the desired sequence.Proof of the theorem. We need the following fact [9]: every nonquasireflexive Banachspace X contains a bounded away from 0 basic sequence {zn}∞n=0 for which ||zn|| ≤1and the set1

{||kXi=jzi(i+1)/2+j||}∞j=0,∞k=jis bounded. Let us denote by Z the closure of the linear span of the sequence {zn}∞n=0 ⊂X.

In order to prove the theorem it is sufficient by Lemma 1 to find a subspace Γ ⊂Z∗for which Γ(α) ̸= Γ(α+1) = Z∗.Let us introduce some notations. We will write zji for z(j+i−1)(j+i)/2+j, biorthogonalfunctionals of the system zn(zji ) will be denoted by ˜zn(˜zji ).

By abovementioned resultfrom [9] we havesupj,m ||mXi=1zji || = M1 < ∞.Therefore for every j = 0, 1, 2, . .

. the sequence {Pmi=1 zji }∞m=1 has at least one w∗-limitpoint in Z∗∗.

Let us for every j = 0, 1, 2 . .

. choose one of such limit points and denoteit by fj.

It is clear that ||fj|| ≤M1.Lemma 2. For every vector g0 ∈Z∗∗of the form afj + zrs(a > 0, r ̸= j), everycountable ordinal α and every infinite subset A ⊂N such that j, r ̸∈A there exists acountable subset Ω(g0, α, A) ⊂Z∗∗such that1) The set K(g0, α, A) defined by K(g0, α, A) = (∩{kerh : h ∈Ω(g0, α, A)}) satisfiesthe condition: (K(g0, α, A))(α) ⊂ker g0.2) All vectors h ∈Ω(g0, α, A) are of the form h = a(h)fj(h) + zr(h)s(h) with j(h), r(h) ∈A ∪{j, r}, a(h) > 0, and if we have j(h) = r or r(h) = r then h = g0.3) Every finite linear combination of vectors {˜zn} which is of the formb˜zrs + u; u ∈lin({˜ztk}∞k=1,t∈A∪{j})(1)must be in (Q(b, g0, α, A)(α) (where Q(b, g0, α, A) is the set of all linear combinations ofthe type (1) which are in K(g0, α, A)) if it is in ker g0.

(Here we need to remark thatthe definition of the w∗-derived sets for subsets which are not subspaces is the same. )At first we will finish the proof of the theorem with the help of lemma 2.

¿From 1) itfollows that (K(g0, α, A))(α) ̸= Z∗. Let us show that 3) implies (K(g0, α, A))(α+1) = Z∗.Let us recall that {zn}∞n=0 is a basis in Z, therefore {˜zn}∞n=0 is a w∗-Schauder basis in Z∗[10, p. 155], i.e.

every vector ˜z ∈Z∗can be represented as ˜z = w∗−limn→∞Pnk=1 ak˜zk,where ak = ˜z(zk).It is clear that vectors ˜yn = Pnk=0 ak˜zk −g0(Pnk=0 ak˜zk)˜zjn/a can be represented inform ˜y1n + ˜y2n where ˜y1n is of type (1) and is in ker g0, and ˜y2n is a finite linear combinationof the vectors {˜zn}∞n=0\({˜ztk}∞k=1,t∈A∪{j} ∪{˜zrs}), and, hence, ˜y2n ∈K(g0, α, A). By 3)we have ˜y1n ∈(K(g0, α, A))(α).

Therefore ˜yn ∈(K(g0, α, A))(α). It is clear also thatw∗−lim ˜yn = ˜z.

Thus the proof of the theorem is complete.Proof of Lemma 2. A.

Let us suppose that the assertion of Lemma 2 is true for ordinalα. Let us show that it is true for α + 1.

Let us represent the set A as a countable unionof pairwise disjoint infinite subsets, A = ∪∞k=0Ak. Let εi > 0 (i ∈N) be such thatP∞i=1 εi < ∞.

Let us introduce the sequence {gn}∞n=1 ⊂Z∗∗by gn = εnfp(n) + zjn,where p : N →A0 is some one-to-one mapping. Let Ω(gn, α, An) be the sets whose2

existence follows from our assumption. Let us show that we can let Ω(g0, α + 1, A) =(∪∞n=1Ω(gn, α, An)) ∪{g0}.

It is clear that 2) is satisfied.Let us show that 1) is satisfied. In order to do this we will using transfinite inductionshow that for β ≤α + 1 we have (K(g0, α + 1, A))(β) ⊂ker g0.

Let us suppose thatwe prove this for certain β ≤α. Let us derive from this assumption that (K(g0, α +1, A))(β+1) ⊂ker g0.

In order to do this we must prove that for any w∗-convergentsequence {˜yi}∞i=1 ⊂(K(g0, α + 1, A))(β) its limit ˜y = w∗−lim ˜yi belongs to ker g0. Wehave already noted that {˜zn} is w∗-basis in Z∗.

Let us estimate coefficients {α(i)jn}∞n=1which are staying near vectors {˜zjn}∞n=1 in the corresponding w∗-decomposition of ˜yi. Itis clear that sup ||˜yi|| = M2 < ∞.

¿From β ≤α and induction hypothesis we obtaingn(˜yi) = 0. We have gn(˜yi) = α(i)jn + εnfp(n)(˜yi), hence|α(i)jn| ≤εnM1M2(n ∈N).

(2)Therefore vector ˜yi can be represented in the form ˜yi = ui + vi in such a way that w∗-decomposition of ui doesn‘t involve vectors ˜zrs and {˜zjn}∞n=1 and vi = ai˜zrs +P∞n=1 α(i)jn˜zjn.By (2) this series is strongly convergent. It is clear that w∗-convergence implies coordi-natewise convergence, therefore it follows from (2) that ˜y also can be represented in theform ˜y = u + v in the same manner.

So we have v = w∗−lim vi and u = w∗−lim ui.From ui(Pmk=1 γkzjk + δzrs) = 0 (i ∈N) and analogous equality for u which holds forall m, {γk}mk=1 and δ it follows that g0(ui) = 0, g0(u) = 0. Since ˜yi ∈ker g0 we havevi ∈ker g0.Thus it is sufficient to prove that v ∈ker g0.This assertion follows from v =w∗−lim vi; vi ∈ker g0 and the following two facts:a) For every i ∈N we have vi ∈V := {˜z ∈Z∗: ˜z = P∞k=1 ck˜zjk + d˜zrs, |ck| ≤εkM1M2, |d| ≤M2} and the set V is compact in the strong topology.b) Initial topology of compact space coincides with any other Hausdorfftopologywhich is weaker than it [11, p. 249].Therefore we proved that if β ≤α and (K(g0, α + 1, A))(β) ⊂ker g0 then(K(g0, α + 1, A))(β+1) ⊂ker g0.The case of limit ordinal is evident.The proof of 3) follows immediately from the induction hypothesis and the followingassertions, which can be easily verified:(i) any vector of type (1) can be represented in form b˜zrs + Pmk=1 ak˜zjk + uk, whereuk ∈lin ({˜ztl}∞l=1,t∈Ak∪{p(k)}).

(ii) if M contains all vectors of the type (1) from ker g0 then M(1) contains all vectorsof the type (1). (iii) if collection {ak}mk=1 ⊂R and b ∈R are such that g0(b˜zrs + Pmk=1 ak˜zjk) = 0 thenwe have Q(b, g0, α + 1, A) ⊃b˜zrs +Pmk=1 Q(ak, gk, α, Ak).

(iv) for any finite collection of sets Mi ⊂Z∗, i = 1, 2, . .

. , m, and any ordinal α wehave (Pmi=1 Mi)(α) ⊃Pmi=1(Mi)(α).B.

The case in which α is a limit ordinal. By [11, p. 72] we can find an increasingsequence {αi}∞i=1 of ordinals for which α is a limit.3

Let {An}∞n=0 and {gn}∞n=1 are the same as in A. By induction hypothesis we can findsubsets Ω(gn, αn, An) ⊂Z∗∗.

The fact that (∪∞n=1Ω(gn, αn, An))∪{g0} is the desired setcan be proved in the same way as in A with unique distinction that inequalities (2) arevalid not for all n but for all exept finite number of them.Received January 19.1987.Address: Institute for low Temperature Physics andEngineeringUkrainian Academy of Sciences47 Lenin avenueKharkov 310164 UkraineUSSR.REFERENCES1. Mazurkiewicz S. Sur la derivee faible d‘un ensemble de fonctionnelles lineaires,Stud.

Math. 2.

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1(Warszawa, 1932).3. McGehee O.C.

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Bases in Banach spaces, I (Berlin, Springer, 1970).11. Aleksandrov P.S.

Introduction to the set theory and general topology (Moscow,Nauka, 1977).COMMENTARY BY THE AUTHOR (July, 1991).I would like to say several words about history and applications of “long chains” ofw∗-derived sets.S.Mazurkiewicz [1] was the first who found a subspace Γ ⊂(c0)∗for which Γ(1) ̸= Γ(2).S. Banach [2, p. 209] developed this construction and for any n ∈N found a subspaceΓ ⊂(c0)∗for which Γ(n) ̸= Γ(n+1) and stated analogous result for any countable ordinalα.

But S.Banach’s proof of this result never appeared. This statement has been provedby O.C.McGehee [3].

(It is interesting to remark that O.C.McGehee used results fromFourier analysis.) At almost the same time D.Sarason (using the complex analysis)obtained analogous results for l1 and L1/H1 [4, 5].

In [6] B.V.Godun observed that thelength of the chain of different w∗-derived sets must be countable and cannot be equal4

to a limit ordinal. In [7] B.V.Godun proved that for any nonquasireflexive separableBanach space X and any n ∈N there exists a total subspace Γ ⊂X∗for whichΓ(n) ̸= X∗.The problem of existence of a total subspace Γ ⊂X∗for which Γ(n) ̸= X∗for alln ∈N is turned out to be important in the theory of topological vector spaces [12].

Thisproblem has been posed by V.B.Moscatelli at the Ninth Seminar (Poland - GDR) onOperator Ideals and Geometry of Banach spaces (Georgenthal, April, 1986) [13, Prob-lem 17]. This problem is solved by the result of present paper (At the time of writingit the author had not any information about [12] and [13]) Independently and almostat the same time V.B.Moscatelli [14] proved that for any separable nonquasireflexiveBanach space X there exists a subspace Γ ⊂X∗for which Γ(ω) ̸= X∗(where ω is thefirst infinite ordinal).

In [15] V.B.Moscatelli obtained this result in more explicit form.The reader may consult [12, 15, 16] for applications of the “long chains” in the theory oftopological vector spaces. In [17] “long chains” are used to answer Kalton’s question onuniversal biorthogonal systems.

There are also certain applications of “long chains” inthe theory of ill-posed problems, or, more precisely, in the problems of regularizabilityof inverses of injective continuous linear operators (see [8, 18, 19]).Some connections of “long chains” with harmonic analysis are discussed in [20].12. Dierolf S., Moscatelli V.B.

A note on quojections, Funct. et Approxim.

(1987),n. 17, 131-138.13. Open Problems, Presented at the Ninth Seminar (Poland - GDR) on OperatorIdeals and Geometry of Banach Spaces, Georgenthal, April, 1986, Forschung.

Friedrich- Schiller - Universitat, Jena, N/87/28, 1987.14. Moscatelli V.B.

On strongly non-norming subspaces, Note Mat. 7 (1987), 311-314.15.Moscatelli V.B.

Strongly nonnorming subspaces and prequojections, StudiaMath. 95 (1990), 249-254.16.

Metafune G., Moscatelli V.B. Quojections and prequojections, in: Advancesin the Theory of Frechet Spaces, ed.by T.Terzioglu, Kluwer Academic Publishers,Dordrecht, 1989, 235-254.17.

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84 (1986), 25–37.18. Ostrovskii M.I.

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5, 3403–3410.19. Ostrovskii M.I., Regularizability of superpositions of inverse linear operators,Teor.

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55 (1991), 96–100 (in Russian). Engl.transl.

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1, 652–655.20. Katznelson Y., McGehee O.C.

Some sets obeying harmonic synthesis, Israel J.Math. 23 (1976), no.

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