THE SZLENK INDEX OF L (X) AND A p p RYAN M. CAUSEY Abstract. Given a Banach space X, a w∗-compact subset of X∗, and 1 < p < ∞, we 7 provide an optimal relationship between the Szlenk index of K and the Szlenk index of 1 an associated subset of L (X)∗. As an application, given a Banach space X, we prove an p 0 optimal estimate of the Szlenk index of L (X) in terms of the Szlenk index of X. This 2 p extends a result of Ha´jek and Schlumprecht to uncountable ordinals. More generally, given n a an operator A : X → Y, we provide an estimate of the Szlenk index of the “pointwise A” J operator A :L (X)→L (Y) in terms of the Szlenk index of A. p p p 2 2 ] A 1. Introduction F . h Throughout this work, X will be a fixed Banach space and K ⊂ X∗ will be a w∗-compact, t a non-empty subset. For1 < p < ∞, welet K denote thew∗-closureinL (X)∗ ofall functions m p p of the form gh ∈ L (X∗) ⊂ L (X)∗, where g : [0,1] → K is simple and Lebesgue measurable, [ q p 1 and h ∈ B . Recall that these functions act on L (X) by hgh,fi = hg(̟),f(̟)ih(̟)d̟ 1 Lq p 0 6v for f ∈ Lp(X). Note that if R > 0 is such that K ⊂ RBX∗, Kp ⊂ RRBLp(X)∗, so that Kp is 2 also w∗-compact. If K = BX∗, Kp = BLp(X)∗ by the Hahn-Banach theorem. If A : X → Y 2 is an operator, then there exists a “pointwise A” operator A : L (X) → L (Y) given by 6 p p p 0 (Apf)(̟) = A(f(̟)) for all ̟ ∈ [0,1]. Then if K = A∗BY∗, Kp = (Ap)∗BLp(Y)∗, which . 1 follows from the Hahn-Banach theorem. Thus it is natural to examine what relationship 0 7 exists between K and Kp. In particular, one may ask what relationship exists between the 1 Szlenk indices ofthese sets. To thatend, weobtaintheoptimal relationship. Inwhat follows, : v ω denotes the first infinite ordinal. i X r Theorem 1. Fix 1 < p < ∞. Suppose that ξ is an ordinal such that Sz(K) 6 ωξ. Then a Sz(K ) 6 ω1+ξ. If K is convex, Sz(K ) 6 ωSz(K). If K is convex and Sz(K) > ωω, p p Sz(K) = Sz(K ). p Using the facts stated in the introduction that Kp = (Ap)∗BLp(Y)∗ if K = A∗BY∗, we immediately deduce the following from Theorem 1. Corollary 2. Fix 1 < p < ∞. If A : X → Y is an operator and K = A∗BY∗, then Sz(A ) 6 ωSz(A), and if Sz(A) > ωω, Sz(A ) = Sz(A). In particular, A is Asplund if p p p and only if A is. Applying Corollary 2 to the identity of a Banach space, we extend the result of Hájek and Schlumprecht from [8] to uncountable ordinals. 1 2 RYANM. CAUSEY We recall that K is said to be w∗-fragmentable if for any non-empty subset L of K and any ε > 0, there exists a w∗-open subset U of X∗ such that L∩U 6= ∅ and diam(L∩U) < ε. We recall that K is w∗-dentable if for any non-empty subset L of K and any ε > 0, there exists a w∗-open slice S of X∗ such that L∩U 6= ∅ and diam(L∩S) < ε. We recall that a w∗-open slice is a subset of X∗ of the form {x∗ ∈ X∗ : Re x∗(x) > a} for some x ∈ X and a ∈ R. As mentioned in [5], a consequence of Corollary 2 is that if Sz(K) 6 ωξ, then Sz(K) 6 Dz(K) 6 ω1+ξ, where Dz(K) denotes the w∗-dentability index of K. Thus Corollary 2 implies that K is w∗-dentable if and only if it is w∗-fragmentable. In addition to considering the Szlenk index of a set, one may consider the ξ-Szlenk power type p (L) of the set L, which is important in ξ-asymptotically uniformly smooth renorm- ξ ings of Banach spaces and operators. The concept of a ξ-asymptotically uniformly smooth operator was introduced in [6], and further sharp renorming results regarding the ξ-Szlenk power type of an operator were established in [4]. To that end, we have the following. Theorem 3. For any ordinal ξ and any 1 < p < ∞, if 1/p + 1/q = 1, p (K ) 6 1+ξ p max{q,p (K)}. ξ In the case that ξ > ω and p (K) 6 p, p (K) = p (K ), in showing that Theorem 3 is ξ ξ ξ p sharp in some cases. The author wishes to thank P.A.H. Brooker, P. Hájek, N. Holt, and Th. Schlumprecht for helpful remarks during the preparation of this work. 2. L (X), Trees, Szlenk index, games p 2.1. Trees, Γ , P , and stablization results. Given a set Λ, we let Λ 0, there exist x ,...,x ∈ M and a level preserving extended pruning 1 n (σ,τ) : Γ .X → Γ .X such that for every 1 6 i 6 n and every Λ ∋ s (cid:22) t ∈ ξ,n ξ,n ξ,n,i MAX(Γ .X), d(x ,f(σ(s),τ(t))) < δ. ξ,n i 4 RYANM. CAUSEY (iii) If F is a finite set and f : MAX(Γ .X) → F is any function, there exists a level ξ,n preserving extended pruning (σ,τ) : Γ .X → Γ .X such that f ◦ τ| is ξ,n ξ,n MAX(Γξ,n.X) constant. (iv) For any natural numbers k < ... < k 6 n, there exists an extended pruning (σ,τ) : 1 r Γ .X → Γ .X such that for every 1 6 i 6 r, σ(Λ ) ⊂ Λ . ξ,r ξ,n ξ,n,i ξ,n,ki 2.2. The Szlenk index, Szlenk power type. Given a w∗-compact subset L of X∗ and ε > 0, we let s (K) denote the set consisting of those x∗ ∈ L such that for every w∗- ε neighborhood V of x∗, diam(L∩V) > ε. We define the transfinite derivations s0(L) = L, ε sξ+1(L) = s (sξ(L)), ε ε ε and if ξ is a limit ordinal, sξ(L) = sζ(L). ε ε \ ζ<ξ If there exists an ordinal ξ such that sξ(L) = ∅, we let Sz(L,ε) be the minimum such ε ordinal. Otherwise we write Sz(L,ε) = ∞. Since sξ(L) is w∗-compact, we deduce that ε Sz(L,ε) cannot be a limit ordinal. We agree to the conventions that ω∞ = ∞ and ξ < ∞ for any ordinal ξ. We let Sz(L) = sup Sz(L,ε). If B : Z → W is an operator, we ε>0 let Sz(B,ε) = Sz(B∗BW∗,ε), Sz(B) = Sz(B∗BW∗). If Z is a Banach space, Sz(Z,ε) = Sz(I ,ε) and Sz(Z) = Sz(I ). Z Z We recall that a set L ⊂ X∗ is called w∗-fragmentable if for any ε > 0 and any w∗- compact, non-empty subset M of L, s (M) ( M. This is equivalent to Sz(L) < ∞. We say ε an operator B : Z → W is Asplund if B∗BW∗ is w∗-fragmentable, which happens if and only if Sz(B) < ∞. We say a Banach space Z is Asplund if I is Asplund. These are not the Z original definitions of Asplund spaces and operators, but they are equivalent to the original definitions (see [1]). If Sz(K) 6 ωξ+1, then for any ε > 0, Sz(K,ε) 6 ωξn for some n ∈ N. We let Sz (K,ε) ξ be the smallest n ∈ N such that Sz(K,ε) 6 ωξn. We define the ξ-Szlenk power type p (K) ξ of K by logSz (K,ε) p (K) = limsup ξ . ξ |log(ε)| ε→0+ This value need not be finite. By convention, we let p (K) = ∞ if Sz(K) > ωξ+1. We ξ let pξ(A) = pξ(A∗BY∗) and pξ(X) = pξ(BX∗). The quantities pξ(X), pξ(A) are important for the renorming theorem of ξ-asymptotically uniformly smooth norms with power type modulus. Given a w∗-compact subset L of X∗ and ε > 0, we let HL denote the set of Cartesian ε products n C such that C ∈ C for each 1 6 i 6 n and such that there exist (x )n ∈ i=1 i i i i=1 n C anQd x∗ ∈ K such that for each 1 6 i 6 n, Re x∗(x ) > ε. i=1 i i Q THE SZLENK INDEX OF L (X) AND A 5 p p 2.3. The Szlenk index of K . Recall that for 1 < p < ∞, L (X) denotes the space of p p equivalence classes of Bochner integrable functions f : [0,1] → X such that kfkp < ∞, where [0,1] is endowed with its Lebesgue measure. Recall also that if 1 < q < ∞R , L (X∗) is q isometrically included in L (X)∗ by the action p f 7→ hg,fi, Z for g ∈ L (X∗). We also recall that if ̺ : X → R is any Lipschitz function, then for any q f ∈ L (X), ̺◦f ∈ L . p p We note that the Szlenk index and the ξ Szlenk power type of K are unchanged by scaling K by a positive scalar or by replacing K with its balanced hull. Moreover, for a positive scalar c, (cK) = cK , which has the same Szlenk index and ξ-Szlenk power type as K . If p p p TK is the balanced hull of K, K ⊂ (TK) and Sz(K) = Sz(TK) ([3, Lemma 2.2]) so that p p Theorem 1, Corollary 2, and Theorem 3 hold in general if they hold under the assumption that K ⊂ BX∗ is balanced. Therefore we can and do assume throughout that K ⊂ BX∗ and K is balanced. Let ̺ : X → R be given by ̺(x) = maxx∗∈K Re x∗(x). Since we have assumed K is balanced, ̺(x) = maxx∗∈K |x∗(x)|. It is easy to see that for any 1 < p < ∞ and any f ∈ Lp(X), k̺(f)kLp = maxf∗∈Kp Re f∗(f). Combining this fact with [4, Corollary 2.4] and the proof of that corollary, we obtain the following. Theorem 5. Fix 1 < p,α < ∞. (i) If for every B-tree T with o(T) = ω1+ξ and every normally weakly null (f ) ⊂ t t∈T.Lp(X) B , Lp(X) inf k̺(f)k : t ∈ T.L (X),f ∈ co(f : ∅ ≺ s (cid:22) t) = 0, Lp p s (cid:8) (cid:9) then Sz(K ) 6 ω1+ξ. p (ii) If there exists a constant C such that for every n ∈ N, every B-tree T with o(T) = ω1+ξn, and every normally weakly null collection (f ) ⊂ B , t t∈T.Lp(X) Lp(X) inf k̺(f)k : t ∈ T.L (X),f ∈ co(f : ∅ ≺ s (cid:22) t) 6 Cn−1/α, Lp p s (cid:8) (cid:9) then p (K ) 6 α. 1+ξ p Proposition 6. Suppose T is a non-empty B-tree. Suppose also that (C ) ⊂ C is fixed s s∈T.X and for s = (ζ ,Z )k ∈ T.X, let λ(s) = Z ∩ C . Suppose that S is a non-empty, well- i i i=1 k s founded B-tree and θ : S.X → T.X is a pruning. For s ∈ S.X, let s(s) = |s| λ(θ(s| )). i=1 i If ε > 0 is such that for every t ∈ S.X, s(s) ∈ HK 6= ∅, then for any 0 Q< δ < ε, any ε 0 6 γ < o(S), and any s ∈ Sγ.X, s(s) ∈ Hsγδ(K) 6= ∅. Moreover, for any 0 < δ < ε, ε Sz(K,δ) > o(S). Proof. We induct on γ. The base case is the hypothesis. Assume γ+1 < o(S) and the result holds for γ. Assume s ∈ Sγ+1.X, which means there exists ζ such that sa(ζ,Z) ∈ Sγ.X for all Z ∈ codim(X). Then for every Z ∈ codim(X), there exists Z > W ∈ codim(X) Z 6 RYANM. CAUSEY such that s(sa(ζ,Z)) ⊂ s(s) × B . From this and the inductive hypothesis, for every WZ Z ∈ codim(X), we fix x ∈ B , (xZ)|s| ∈ s(s), and x∗ ∈ sγ(K) such that Re x∗(x ) > ε Z WZ i i=1 Z δ Z Z and Re x∗(xZ) > ε for each 1 6 i 6 |s|. By compactness of s(s) × K with the product Z i topology, where λ(θ(s| )) has its norm topology and K has its w∗-topology, i ∅ 6= {(xY,...,xY ,x∗ ) : Z > Y ∈ codim(X)} ⊂ s(s)×K. 1 |s| Y \ Z∈codim(X) Fix (x ,...,x ,x∗) lying in this intersection. Obviously x∗ ∈ sγ(K). Moreover, for any 1 |s| δ w∗-neighborhood V of x∗, there exists Z ∈ codim(X) such that ker(x∗) ⊂ Z and x∗ ∈ V, Z whence diam(sγ(K)∩V) > kx∗ −x∗k > Re (x∗ −x∗)(x ) = Re x∗(x ) > ε > δ. δ Z Z Z Z Z This implies x∗ ∈ sγ+1(K). It is obvious that Re x∗(x ) > ε for all 1 6 i 6 |s|. This shows δ i that s(s) ∈ Hsγδ+1(K) and completes the successor case. ε Finally, assume γ < o(S) is a limit ordinal and the result holds for all ordinals less than γ. Fix s ∈ Sγ.X and let s(s)×K be topologized as in the successor case. By the inductive hypothesis, for all β < γ, there exists (xβ,...,xβ ,x∗) ∈ s(s)×K such that x∗ ∈ sβ(K) and 1 |s| β β ε for all 1 6 i 6 |s|, Re x∗(xβ) > ε. By compactness of |s| λ(θ(s| )) ×K, β i i=1 i (cid:0)Q (cid:1) {(xµ,...,xµ ,x∗) : µ > β} =6 ∅. 1 |s| µ \ β<γ Clearly any (x ,...,x ,x∗) lying in this intersection is such that x∗ ∈ sγ(K) and for any 1 |s| δ 1 6 i 6 |s|, Re x∗(x ) > ε. This shows that s(s) ∈ Hsγδ(K) and completes the induction. i ε We have shown that for any 0 < δ < ε, Sz(K,δ) > o(S). If o(S) is a limit ordinal, we deduce that Sz(K,δ) > o(S) since Sz(K,δ) cannot be a limit ordinal. If o(S) is a successor, say o(S) = ξ +1, then there exists a length 1 sequence (ζ) ∈ Sξ. For every Z ∈ codim(X), s((ζ,Z)) = W ∩C for some W ⊂ Z. The first part of the proof yields that for each Z θ((ζ,Z)) Z ∈ codim(X), there exists x ∈ W ∩C ⊂ W ∩B and some x∗ ∈ sξ(K) such that Z Z θ((ζ,Z)) Z X Z ζ Re x∗(x ) > ε. Arguing as in the successor case, we deduce that any w∗-limit of a subnet Z Z of (x∗) lies in sξ+1(K), whence Sz(K,δ) > ξ +1 = o(S). Z Z∈codim(X) δ (cid:3) 2.4. Games. SupposeT ⊂ Λ ω1+ξn. If ψ is a strategy for Player I for some game on Γ .X.C, then for any 1 < p < ∞, ξ,n any δ > 0, and any normally weakly null (f ) ⊂ B , there exist s = (ζ ,Z )k ∈ t t∈T.Lp(X) Lp(X) i i i=1 MAX(Γ .X), ∅ = t ≺ t ≺ ... ≺ t ∈ T.L (X), g ∈ co(f : t ≺ u (cid:22) t ), h ∈ B , ξ,n 0 1 k p i u i−1 i i Lp(X) and C ∈ C such that for every 1 6 i 6 k, i (i) h is simple, i (ii) range(h ) = C ⊂ B , i i Zi (iii) kg −h k < δ, i i Lp(X) (iv) (ζ ,Z ) = ψ((ζ ,Z ,C )i−1). i i j j j j=1 Remark 10. For a B-tree S on Λ and s ∈ S, we let S(s) denote those non-empty sequences u ∈ Λ ξ. 8 RYANM. CAUSEY Furthermore, another easy induction yields that if (Sξ)ζ = Sξ+ζ, from which it follows that if o(S) > ξ +ζ, o(Sξ) > ζ. Therefore if s ∈ Sξ+ω, o(Sξ(s)) > ω. Proof of Proposition 9. We first note that if Z ∈ codim(X), L (X)/L (Z) is either the zero p p vector space or isomorphic to L , and therefore has Szlenk index not exceeding ω. As p explained in [5], this means that for any B-tree T with o(T) > ω, any δ > 0, and any normally weakly null (f ) ⊂ B , there exist t ∈ T.L (X), g ∈ co(f : ∅ ≺ s (cid:22) t), t t∈T.Lp(X) Lp(X) p s and h ∈ B such that kg −hk < δ. Moreover, by the density of simple functions, Lp(Z) Lp(X) we may assume this h is simple. Let ψ be a strategy for Player I for a game on Γ .X.C. Let T be a B-tree with o(T) = ξ,n ω1+ξn and define γ : Γ .X∪{∅} → [0,ωξn] by letting γ(t) = max{µ 6 ωξn : t ∈ (Γ .X)µ} ξ,n ξ,n for t ∈ Γ .X and γ(∅) = ωξn. Let s = t = ∅. Now assume that for some k ∈ N and ξ,n 0 0 all 1 6 i < k, s ∈ Γ .X, ζ ∈ [0,ωξn], Z ∈ codim(X), t ∈ T.L (X), g ,h ∈ B , and i ξ,n i i i p i i Lp(X) C ∈ C have been chosen such that for all 1 6 i < k, i (i) h is simple, i (ii) s = (ζ ,Z )i , i j j j=1 (iii) t ≺ t ≺ ... ≺ t , 0 1 k−1 (iv) t ∈ (T.L (X))ωγ(si), i p (v) (ζ ,Z ) = ψ((ζ ,Z ,C )i−1), i i j j j j=1 (vi) g ∈ co(f : t ≺ u (cid:22) t ), i u i−1 i (vii) kg −h k < δ, i i Lp(X) (viii) range(h ) = C ⊂ B . i i Zi If s is maximal in Γ .X, we let s = s , and one easily checks that the conclusions k−1 ξ,n k−1 are satisfied. Otherwise let (ζ ,Z ) = ψ((ζ ,Z ,C )k−1) and s = sa (ζ ,Z ). Let u be k k j j j j=1 k k−1 k k k−1 the sequence of first members of the pairs of t and let U denote the proper extensions of k−1 uk−1 in Tωγ(sk). Then (ftak−1u)u∈U.Lp(X) ⊂ BLp(X) is normally weakly null and o(U) > ω by the remark preceding the proof, so that the previous paragraph yields the existence of some u′ ∈ U.L (X), g ∈ co(f : t ≺ u (cid:22) ta u′), and some simple function h ∈ L (Z ) such p k u k−1 k−1 k p k that kg −h k < δ. Let t = ta u′. In order to apply the remark before the proof, we k k Lp(X) k k−1 note that since s ≺ s , γ(s ) > γ(s )+1. Since k−1 k k−1 k ωγ(s ) > ω(γ(s )+1) = ωγ(s )+ω, k−1 k k the remark preceding the proof applies. Note that C := range(h ) ⊂ B . This completes k k Zk the recursive construction. Since Γ .X is well-founded, eventually this process terminates. ξ,n The resulting s = (ζ ,Z )k ∈ MAX(Γ .X) clearly satisfies the conclusions. i i i=1 ξ,n (cid:3) 3. Definition of an associated space and two games 3.1. The associated space and its properties. If E is a vector space with seminorm k·k, we say a sequence (e )n in E is 1-unconditional provided that for any scalars (a )n i i=1 i i=1 THE SZLENK INDEX OF L (X) AND A 9 p p and any (ε )n ∈ {±1}n, k n ε a e k = k n a e k. Recall that for 1 < p < ∞, a vector i i=1 i=1 i i i i=1 i i space E with seminorm k·Pk which is spannPed by the 1-unconditional basis (e )n is called i i=1 p-concave provided there exists a constant C such that for any (f )n ⊂ L , i i=1 p n n k f e k 6 Ck kf k e k . i i Lp(E) i Lp i E X X i=1 i=1 The smallest such constant C is denoted by M (E). (p) Given x ∈ span(e : 1 6 i 6 n), where (e )n is a Hamel basis for the seminormed i i i=1 space E, we write x = n a e and supp(x) = {i 6 n : a 6= 0}. We say the vectors i=1 i i i x ,...,x ∈ span(e : 1 6Pi 6 n) are disjointly supported if the sets supp(x ), ..., supp(x ) 1 n i 1 n are pairwise disjoint. For 1 < β < ∞, we say that an unconditional Hamel basis (e )n for a seminormed space i i=1 E satisfies an 1-lower ℓ estimate provided that for any m ∈ N and any disjointly supported β elements (x )m ⊂ E, i i=1 m m kx kβ 1/β 6 k x k. i i (cid:0)Xi=1 (cid:1) Xi=1 Theorem 11. [7, Theorem 1.f.7] Fix 1 < β < p < ∞. There exists a constant C′ = C′(β,p) such that if (e )n is a 1-unconditional basis for the seminormed space E which satisfies a i i=1 1-lower ℓ estimate, then E is p-concave and M (E) 6 C′. β (p) For the remainder of this section, T is a fixed, non-empty B-tree. Fora non-empty set J, we let c (J)bethespan ofthecanonical Hamel basis (e ) inthe 00 j j∈J space of scalar-valued functions on J, where e is the indicator of the singleton {j}. We let e∗ j j denote the coordinate functional to e . Given x ∈ c (J), we may write x = a e . Then j 00 j∈J j j we define |x| to be |a |e . A suppression projection is an operator P frPom span(e∗ : j ∈ j∈J j j j J) into itself such tPhat there exists a subset F of J such that P a e∗ = a e∗. j∈J j j j∈F j j For 0 < φ < θ < 1, let P P k N = {0}∪ θ e∗ : t = (ζ ,Z ,C )|t| ∈ T.X.C,1 6 j < ... < j 6 |t|, θ,φ,T n X t|ji i i i i=1 1 k i=1 k Z ∩C ∈ HK ⊂ span(e∗ : t ∈ T.X.C). Y ji ji φ o t i=1 For 0 < φ < θ < 1 and 1 < α < ∞, let k k M = a g :g ∈ ∪∞ N ,a > 0, aα 6 1,supp(g ) are pairwise disjoint . θ,φ,α,T i i i n=1 θn,φn,T i i i nX X o i=1 i=1 Note that the set M is closed under suppression projections. θ,φ,α,T We define the seminorm k·k on c (T.X.C) by θ,φ,α,T 00 kxk = sup{f(|x|) : f ∈ M }. θ,φ,α,T θ,φ,α,T 10 RYANM. CAUSEY Claim 12. Fix 1 < α < ∞ and 0 < φ < θ < 1. For any t ∈ T.X.C, (e )|t| is 1- t|i i=1 unconditional and satisfies a 1-lower ℓ estimate in its span, where 1/α+1/β = 1. β Proof. Note that 1-unconditionality is obvious. Fix x ,...,x ∈ span(e : 1 6 i 6 |t|) 1 n t|i with disjoint supports. That is, there exist pairwise disjoint subsets S ,...,S of {1,...,|t|} 1 n such that x ∈ span(e : j ∈ S ). Then there exist g ,...,g ∈ M such that for each i t|j i 1 n θ,φ,α,T 1 6 i 6 n, g (|x |) = kx k . Since M is closed under suppression projections, we i i i θ,φ,α,T θ,φ,α,T mayassume thatsupp(g ) ⊂ S foreach1 6 i 6 n. Thenif(a )n aresuch that n aα = 1, i i i i=1 i=1 i a > 0, and n a kx k = ( n kx kβ )1/β, g := n a g ∈ M Pand i i=1 i i θ,φ,α,T i=1 i θ,φ,α,T i=1 i i θ,φ,α,T P P P n n n n k x k > g x = a g (|x |) = kx kβ )1/β. i θ,φ,α,T i i i i i θ,φ,α,T Xi=1 (cid:0)(cid:12)Xi=1 (cid:12)(cid:1) Xi=1 (cid:0)Xi=1 (cid:12) (cid:12) (cid:3) Claim 13. Fix 1 < α < ∞ and 0 < φ < θ < 1. For any t = (ζ ,Z ,C )k ∈ T.X.C, any i i i i=1 sequence (x )k ∈ k Z ∩C , and any sequence (a )k of non-negative scalars, i i=1 i=1 i i i i=1 Q k k 1 ̺( a x ) 6 k a e k . i i θ −φ i t|i θ,φ,α,T X X i=1 i=1 Proof. We recall that if C ∈ C, C ⊂ B by the definition of C. With t, (x )k ∈ k Z ∩C , X i i=1 i=1 i i and (a )k as in the statement, fix x∗ ∈ K such that Re x∗( k a x ) = ̺Q( k a x ). i i=1 i=1 i i i=1 i i For all j ∈ N, let B = {i 6 k : Re x∗(x ) ∈ (φj,φj−1]}. NPote that for evePry j ∈ N, j i θj e∗ ∈ N , so i∈Bj t|i θj,φj,T P k k φj−1 a = φ−1(φ/θ)j(θj e∗ )( a e ) 6 φ−1(φ/θ)jk a e k . i t|i i t|i i t|i θ,φ,α,T X X X X i∈Bj i∈Bj i=1 i=1 Then k ∞ ∞ ∞ k ̺( a x ) 6 a Re x∗(x ) 6 φj−1 a 6 φ−1(φ/θ)jk a e k i i i i i i t|i θ,φ,α,T X XX X X X X i=1 j=1 i∈Bj j=1 i∈Bj j=1 i=1 k 1 = k a e k . θ−φ i t|i θ,φ,α,T X i=1 (cid:3) Corollary 14. Fix 1 < p,α,β < ∞ with 1/α+1/β = 1 and β < p. Let C′ = C′(β,p) be the constant from Theorem 11. Suppose that ξ is an ordinal, n is a natural number, ε > 0, and 0 < φ < θ < 1 are such that Player I has a winning strategy in the game with target set t ∈ MAX(Γ .X.C) : k P (s)e k > ε . n ξ,n X ξ,n s θ,φ,α,Γξ,n o s(cid:22)t

The Szlenk index of $L_p(X)$ and $A_p$

Similar The Szlenk index of $L_p(X)$ and $A_p$