aa r X i v : . [ m a t h . F A ] A p r USING BOUNDARIES TO FIND SMOOTH NORMS
VICTOR BIBLE
Abstract.
The aim of this paper is to present a tool used to showthat certain Banach spaces can be endowed with C k smooth equiv-alent norms. The hypothesis uses particular countable decomposi-tions of certain subsets of B X ∗ , namely boundaries. Of interest isthat the main result unifies two quite well known results. In thefinal section, some new corollaries are given. Introduction
We say the norm of a Banach space ( X, || · || ) is C k smooth if its k thFr´echet derivative exists and is continuous at every point of X \ { } .The norm is C ∞ smooth if this holds for all k ∈ N . This paper isconcerned with the problem of establishing sufficient conditions forwhen a Banach space has a C k smooth renorming, for k ∈ N ∪ {∞} . Definition 1.1.
A subset B ⊆ B X ∗ is a called a boundary if for each x in the unit sphere S X , there exists f ∈ B such that f ( x ) = 1. Example 1.2.
The following will be boundaries for any Banach space X . (1) The dual unit sphere S X ∗ . This is a consequence of the Hahn-Banach Theorem.(2) The set of extreme points of the dual unit ball, Ext( B X ∗ ). Thisfollows from the proof of the Krein-Milman Theorem ([3, Fact3.45]). Date : June 28, 2018.
Key words and phrases. boundaries, C k smooth norms, countable covers, injec-tive tensor product, Orlicz, renormings.The author is supported financially by Science Foundation Ireland under GrantNumber ‘SFI 11/RFP.1/MTH/3112’. Given ε > || · || and ||| · ||| on a Banach space X , say ||| · ||| ε - approximates || · || if, for all x ∈ X ,(1 − ε ) || x || ≤ ||| x ||| ≤ (1 + ε ) || x || . The notion of a boundary plays an important role in this area of study.Frequently, the existence of a boundary with certain properties givesrise to the desired renormings, as seen in the following result of H´ajek,which is part of a more general theorem.
Theorem 1.3 ([7, Theorem 1]) . If ( X, || · || ) admits a boundary con-tained in a || · || - σ -compact subset of B X ∗ , then X admits an equivalent C ∞ smooth norm that ε -approximates || · || . H´ajek and Haydon provided another sufficient condition for whenthis property holds, namely when X = C ( K ) and K is a compactHausdorff σ -discrete space. We call a topological space K σ -discrete if K = S ∞ n =0 D n , where each D n is relatively discrete: given x ∈ D n ,there exists U x open in K such that U x ∩ D n = { x } . Theorem 1.4 ([9, Theorem 5.1]) . Let K be a σ -discrete compact space.Then, given ε > , C ( K ) admits an equivalent C ∞ smooth norm that ε -approximates || · || ∞ . It is worth remarking that, in certain cases, these conclusions havebeen strengthened. In [2], it is shown that if X has a countable bound-ary then X has an equivalent analytic norm which ε -approximates theoriginal norm. Moreover, if C ( K ) admits an analytic renorming, then K is countable [8]. For a norm || · || to be analytic we mean it is areal valued analytic function on X \ { } . Analytic functions on Banachspaces are defined and explored [14].The Orlicz functions M for which the corresponding Orlicz sequencespaces l M and Orlicz function spaces l M (0 , , l M (0 , ∞ ) have an equiv-alent C ∞ smooth norm were characterised in [12]. Futhermore, theOrlicz sequence spaces h M with equivalent analytic norm were charac-terised in [10].The main result of this paper, Theorem 2.1, generalises these resultsas corollaries. It also takes into account smoothness of injective tensor SING BOUNDARIES TO FIND SMOOTH NORMS 3 products, in a manner similar to that of [11]. As in the proof of [9,Theorem 5.1], the proof of Theorem 2.1 makes use of two lemmas ([9,Lemma 5.2 and Lemma 5.3]) concerning the so-called generalised Orlicznorm, denoted by || · || φ . The first lemma provides a condition where || · || φ is equivalent to || · || . Definition 1.5.
Let B be a set. Suppose for every element t ∈ B there exists a convex function φ t on [0 , ∞ ) with φ t (0) = 0 andlim α →∞ φ t ( α ) = ∞ (such functions are called Orlicz functions ). Define || · || φ on l ∞ ( B ) by || f || φ = inf ( ρ > X t ∈ B φ t (cid:18) | f ( t ) | ρ (cid:19) ≤ ) . and define ℓ φ ( B ) as the set of f ∈ ℓ ∞ ( B ) satisfying || f || φ < ∞ . Lemma 1.6. [9, Lemma 5.2]
Let || · || φ be as in Definition 1.5. Supposethere exist β > α > with the property φ t ( α ) = 0 and φ t ( β ) ≥ forall t ∈ B . Then ℓ φ ( B ) ∼ = ℓ ∞ ( B ) and α || · || φ ≤ || · || ∞ ≤ β || · || φ . We use || · || φ to define another norm on a more general space X ,which we also denote by || · || φ . The second lemma gives a sufficientcondition for when || · || φ on X is C k smooth. It uses the notion of localdependence on finitely many coordinates and generalises [9, Lemma5.3]. Lemma 1.7.
Let || · || φ be as in Lemma 1.6 and let Π : X → ℓ φ ( B ) bean embedding (non-linear in general), where the map x Π( x )( t ) is aseminorm which is C k smooth on the set where it is non-zero, for all t ∈ B . Assume the assignment || x || φ = || Π( x ) || φ defines an equivalentnorm on X . Suppose for each x ∈ X , with || x || φ = 1 , there exists anopen U ⊆ X containing x , and finite F ⊆ B , such that φ t ( | y ( t ) | ) = 0 when y ∈ U and t ∈ B \ F. Finally, assume that each φ t is C ∞ smooth.Then || · || φ is C k smooth on X . As Lemma 1.7 appears in [9, Lemma 5.3], X is taken to be a closedsubspace of ℓ ∞ ( B ) and Π is the identity. The proof uses the fact that VICTOR BIBLE each coordinate map x → | x ( t ) | is C ∞ smooth on the set where it isnon-zero and uses the implicit function theorem to show that || · || φ isalso C ∞ smooth. In our case, each coordinate map is C k smooth onthe set where it is non-zero and the same argument guarantees that || · || φ is C k smooth.The first part of the proof of Theorem 2.1 is concerned with settingup the necessary framework to apply these lemmas. The remainder usesa series of claims to prove they do in fact hold. In the final section,Theorems 1.3 and 1.4 are obtained as corollaries of Theorem 2.1, alongwith some other results and applications.Before proceeding to the statement of Theorem 2.1, a key notion of w ∗ - locally relatively compact sets ( w ∗ -LRC for short) needs to be intro-duced. This property is first studied in [6], in the context of polyhedralnorms. Definition 1.8 ([6, Definition 5]) . Let X be a Banach space. We call E ⊆ X ∗ w ∗ -LRC if given y ∈ E , there exists a w ∗ -open set U such that y ∈ U and E ∩ U ||·|| is norm compact. Example 1.9 ([6, Example 6]) . The following sets are w ∗ -LRC.(1) Any norm compact or w ∗ -relatively discrete subset of a dualspace.(2) Given X with an unconditional basis ( e i ) i ∈ I and f ∈ X ∗ , definesupp( f ) = { i ∈ I : f ( e i ) = 0 } . Let E ⊂ X ∗ have the property that if f, g ∈ E , then | supp( f ) | = | supp( g ) | < ∞ . E is w ∗ -LRC. Indeed, take f ∈ E and definethe w ∗ -open set U = { g ∈ X ∗ : 0 < | g ( e i ) | < | f ( e i ) | + 1 : i ∈ supp( f ) } . Clearly, if g ∈ U ∩ E , then supp( g ) = supp( f ). Thus U ∩ E is a norm bounded subset of a finite dimensional space. Remark 1.10.
Evidently, w ∗ -LRC sets are preserved under scalar mul-tiplication. Also, the family of σ - w ∗ -LRC subsets of a dual Banachspace forms a σ -ideal. (This is because if E is w ∗ -LRC and F ⊆ E ,then F is w ∗ -LRC. And of course any countable union of σ - w ∗ -LRCsets is again σ - w ∗ -LRC). But in general they do not behave well under SING BOUNDARIES TO FIND SMOOTH NORMS 5 straightforward linear and topological operations. To see this, considerthe following.(1) Let E = { e n : n ∈ N } be the usual basis of c and let F = { } .These sets are both w ∗ -LRC. However, 0 is a w ∗ -accumulationpoint of E and || e n − e m || ∞ = 1 whenever n = m , so E ∪ F isnot w ∗ -LRC.(2) The set E = { δ α +2 − n δ α + n : α < ω is a limit ordinal , n ∈ N } is w ∗ -discrete. But E ||·|| ⊇ { δ α : α < ω is a limit ordinal } . Usingthe fact that the ordinal ω is not σ -discrete, we can see thatthe set { δ α : α < ω is a limit ordinal } , and thus E ||·|| , is not σ - w ∗ -LRC.(3) Consider the space ℓ ⊕ ℓ ( B ℓ ) ≡ ℓ ( N ∪ B ℓ ). Given x ∈ B ℓ ,denote by x its canonical image in ℓ ( N ∪ B ℓ ). Let E = { x ± δ x : x ∈ B ℓ } . This set can be shown to be w ∗ -discrete but E + E ⊇{ x : x ∈ B ℓ } ≡ B ℓ . A conseqeunce of [6, Proposition 12(1)] is that for an infinite dimensional space X , S X ∗ cannotbe covered by a countable union of w ∗ -LRC sets. This resultextends to S Y , where Y is any infinite-dimensional subspace of X ∗ . Because of this, E + E is not σ - w ∗ -LRC.The main result is concerned with renorming injective tensor prod-ucts. Given Banach spaces X and Y , the injective tensor product X ⊗ ε Y is the completion of the algebraic tensor product X ⊗ Y withrespect to the norm || ∞ X i =1 x i ⊗ y i || = sup ( ∞ X i =1 f ( x i ) g ( y i ) : f ∈ B X ∗ , g ∈ B Y ∗ ) . Also note the following facts. If I Y is the identity operator on Y , thengiven f ∈ X ∗ we define f Y = f ⊗ I Y on X ⊗ Y by f Y ( P ∞ i =1 x i ⊗ y i ) = P ∞ i =1 f ( x i ) y i . We have || f Y || = || f || and extend to the completion.Similarly define g X for g ∈ Y ∗ . A useful fact is f ⊗ g = g ◦ f Y = f ◦ g X . Given two boundaries N ⊆ X ∗ and M ⊆ Y ∗ , the set { f ⊗ g : f ∈ N, g ∈ M } is a boundary for X ⊗ ε Y . To see this, take u ∈ X ⊗ ε Y .There exists f ∈ B X ∗ and g ∈ B Y ∗ such that || u || = ( f ⊗ g )( u ) = || g X ( u ) || . Then there exists ˆ f ∈ N such that ˆ f ( g X ( u )) = || u || = VICTOR BIBLE || ˆ f Y ( u ) || . Finally, there exists ˆ g ∈ M such that ˆ g ( ˆ f Y ( u )) = ( ˆ f ⊗ ˆ g )( u ) = || u || . Given a Banach space Y with a C k smooth renorming, Haydon gavea sufficient condition on X for X ⊗ ε Y to have a C k smooth renorming([11, Corollary 1]). This condition involves a type of operator thatare now known as Talagrand operators. Another sufficient condition isgiven in the main result below. It is worth noting that these conditionsare incomparable. For example, the space C [0 , ω ] satisfies Haydon’scondition but not that of Theorem 2.1. On the other hand, if we take K to be the Ciesielski-Pol space as seen in [1], then C ( K ) satisfies thehypothesis of Theorem 2.1 but not Haydon’s condition.2. Main Result
Theorem 2.1.
Let X and Y be Banach spaces and let ( E n ) be a se-quence of w ∗ -LRC subsets of X ∗ , such that E = S ∞ n =0 E n is σ - w ∗ -compact and contains a boundary of X . Suppose further that Y has a C k smooth norm || · || Y for some k ∈ N ∪ {∞} . Then X ⊗ ε Y admits a C k smooth renorming that ε -approximates the canonical injective ten-sor norm. The proof of this theorem is based to some degree on that of [6,Theorem 7]. Given its technical nature, some of that proof is repeatedhere for clarity. We ask the reader to excuse any redundancy.
Proof.
To begin, we can assume E is a boundary and E nw ∗ ⊆ E forall n ∈ N . Indeed, if neccessary, taking E = S ∞ m =0 K m , where K m is w ∗ -compact, we can consider for all n, m ∈ N ,E n ∩ K m ∩ B X ∗ . By [6, Proposition 12 (3)] there exist w ∗ -open sets V n such that if weset A n = E nw ∗ ∩ V n , then E n ⊆ A n ⊆ E n ||·|| and A n is w ∗ -LRC . SING BOUNDARIES TO FIND SMOOTH NORMS 7
Each A n is both norm F σ and norm G δ . So for each n ∈ N , A n \ S k
0. We define ψ : E −→ (1 , ε ) by ψ ( f ) = 1 + 12 ε · − n ( f ) X i ∈ I ( f ) − i . Set ε n = ε · − n . Fix n . As ψ ( L n ) ⊆ (1 , ε ) , there is a finitepartition of L n into sets J, such that diam( ψ ( J )) ≤ ε n . Let P = { I ⊆ J : I is ε n -separated } . This set is non-empty becauseany singleton is in P . For a chain T ⊆ P we have S N ∈ T N ∈ P, so wecan apply Zorn’s Lemma to get Γ ⊆ J, a maximal ε n -separated subsetof J. By maximality, Γ is also an ε n -net. And by the ε n -separation, fora totally bounded set M ⊆ J , the intersection M ∩ Γ is finite.By considering the finite union of these Γ , there exists Γ n ⊆ L n ,with the property that given f ∈ L n there exists h ∈ Γ n so that(1) | ψ ( f ) − ψ ( h ) | ≤ ε n and || f − h || ≤ ε n . Moreover, if M ⊆ L n is totally bounded, M ∩ Γ n is finite. Now define B = S ∞ n =0 Γ n . We are now ready to define || · || φ on ℓ ∞ ( B ).For each f ∈ B we pick a C ∞ Orlicz function φ f so that φ f ( α ) = 0 if α ≤ ψ ( f ) ,φ f ( α ) > α ≥ θ ( f ) , where θ ( f ) = ψ ( f ) − ε n . VICTOR BIBLE
We define || · || φ with respect to these functions, as per Definition1.5. By taking (1 + ε ) − and 1 as the constants in the hypothesis ofLemma 1.6 we have l φ ( B ) ∼ = l ∞ ( B ) and || · || ∞ ≤ || · || φ ≤ (1 + ε ) || · || ∞ . We embed X ⊗ ε Y into ℓ ∞ ( B ) by setting Π( u )( f ) = || f Y ( u ) || Y , f ∈ B . The coordinate map u → || f Y ( u ) || is a seminorm which is C k smooth on the set where it is non-zero for each f ∈ B . Since || Π( u ) || ∞ = || u || , it follows that || · || ≤ || · || φ ≤ (1 + ε ) || · || on X .Suppose for the sake of contradiction that the remaining hypothesisof Lemma 1.7 does not hold. Then we can find u ∈ X ⊗ ε Y with || u || φ = 1, ( u n ) ⊆ X ⊗ ε Y with u n → u and distinct ( f n ) ⊆ B suchthat φ f n ( || f Yn ( u n ) || ) >
0, for all n . Then ψ ( f n ) || f Yn ( u n ) || > n .Take a subsequence of ( f n ), again called ( f n ), such that ψ ( f n ) → α for some α ∈ R . Now take ( g n ) ⊆ S Y ∗ such that || f Yn ( u n ) || = g n ( f Yn ( u n )) . Let ( f, g ) ∈ B X ∗ × B Y ∗ be an accumulation point of ( f n , g n )in the product of the w ∗ -topologies. Then f ⊗ g is a w ∗ -accumulationpoint of ( f n ⊗ g n ) and α ( f ⊗ g )( u ) ≥ α ( f ⊗ g )( u ) < . Case 1: α = 1 . With α = 1, it is evident that α ( f ⊗ g )( u ) =( f ⊗ g )( u ) ≤ || u || . The following claim ensures || u || < Claim 1: If v = 0 , then || v || < || v || φ . Let || v || = 1 and pick p ∈ E , q ∈ S Y ∗ such that 1 = ( p ⊗ q )( v ) . Asnoted above, this is possible because E and S Y ∗ are boundaries of X and Y , respectively. By (1) above, let r ∈ B such that || p − r || ≤ ε n foran appropriate n . Observe that θ ( r )(( r ⊗ q )( v )) ≤ || v || φ holds. Indeed, X l ∈ B φ l (cid:18) || l Y ( v ) || θ ( r )( r ⊗ q )( v ) (cid:19) ≥ φ r (cid:18) || r Y ( v ) || θ ( r ) q ( r Y ( v )) (cid:19) ≥ φ r (cid:18) θ ( r ) (cid:19) > . SING BOUNDARIES TO FIND SMOOTH NORMS 9
Now to prove the claim,1 = ( p ⊗ q )( v )= ( r ⊗ q )( v ) + (( p − r ) ⊗ q )( v )= θ ( r )( r ⊗ q )( v ) + (1 − θ ( r ))( r ⊗ q )( v ) + (( p − r ) ⊗ q )( v ) ≤ || v || φ + (1 − θ ( r ))( r ⊗ q )( v ) + (( p − r ) ⊗ q )( v ) . So we are done if ( θ ( r ) − r ⊗ q )( v ) + (( r − p ) ⊗ q )( v ) > . Indeed, θ ( r ) − ψ ( r ) − ε n − ≥ ε · − n ( r ) − ε n ≥ ε · − n − ε n . Also, ( r ⊗ q )( v ) = r ( q X ( v )) ≥ − || p − r || · || q X ( v ) || ≥ . Thus,( θ ( r ) − r ⊗ q )( v ) + (( r − p ) ⊗ q )( v ) ≥ ε · − n − ε n − ε n = 14 ε · − n − ε n = 14 ε · − n − ε · − n > . And the claim is proven.
Case 2: α > f ∈ E .Fix N large enough so that 1+ ε · − N < (1+ α ) . Because ψ ( f n ) → α we have ψ ( f m ) > (1 + α ) for all m large enough. Hence, n ( f m )
1+ 12 ε · − n ( f ) − (1+ 34 ε · − n ( f n ) ) ≥ ε · − n ( f ) ≥ ε · − m . And if n ( f n ) = n ( f ), then ψ ( f ) − ψ ( f n ) ≥ ε · − n ( f ) X i ∈ I ( f ) − i − X i ∈ I ( f n ) − i = 18 ε · − n ( f ) X i ∈ I ( f ) \ I ( f n ) − i − X i ∈ I ( f n ) \ I ( f ) − i ≥ ε · − n ( f ) − m − X i ∈ J \ I ( f ) − i ≥ ε · − n ( f ) · − m − ≥ ε · − m . SING BOUNDARIES TO FIND SMOOTH NORMS 11
Claim 2d
For h ∈ B, || h ⊗ g || φ ≤ θ ( h ) . If | ( h ⊗ g )( v ) | > θ ( h ) , then X l ∈ B φ l ( || l Y ( v ) || ) ≥ φ h ( || h Y ( v ) || ) ≥ φ h (( h ⊗ g )( v )) > ⇒ || v || φ > . So, || h ⊗ g || φ = sup {| ( h ⊗ g )( v ) | : || v || φ ≤ } ≤ θ ( h ) . We can now prove α ( f ⊗ g )( x ) < . By (1), take h ∈ B such that || f − h || ≤ ε n and | ψ ( f ) − ψ ( h ) | ≤ ε n . We then have α || f ⊗ g || φ ≤ α ( || h ⊗ g || φ + || ( f − h ) ⊗ g || φ ) ≤ α ( || h ⊗ g || φ + || ( f − h ) ⊗ g || ) ≤ α (cid:18) θ ( h ) + ε n (cid:19) . So we are done if α ( θ ( h ) + ε n ) < . Well,1 − αθ ( h ) − αε n > ⇐⇒ θ ( h ) − α − ε n θ ( h ) α > ⇐⇒ ψ ( h ) − ε n − α − ε n θ ( h ) α > . By claim 2c, we have ψ ( h ) − ε n − α ≥ ε n and since θ ( h ) , α <
2, itfollows that ε n θ ( h ) α < ε n .And so, α || f ⊗ g || φ < ⇒ α ( f ⊗ g )( u ) < . (cid:3) Applications
Corollary 3.1.
Suppose X has a σ - w ∗ -LRC and σ - w ∗ -compact bound-ary. Then X has a C ∞ renorming.Proof. Apply Theorem 2.1 to X ⊗ ε R = X . (cid:3) We can now prove Theorems 1.3 and 1.4 as corollaries of Corollary3.1.
Proof of Theorem 1.3.
Any norm compact subset of X ∗ is trivially w ∗ -LRC. The result follows from Corollary 3.1. (cid:3) Proof of Theorem 1.4.
Let K = S ∞ n =0 D n , where each D n is relativelydiscrete. Let δ t be the usual evaluation functionals, δ t ( f ) = f ( t ). Then E n = {± δ t : t ∈ D n } is w ∗ -relatively discrete and so w ∗ -LRC. Moreover, E = S ∞ n =0 E n is a w ∗ -compact boundary of C ( K ) because given any f ∈ C ( K ), there exists t ∈ K such that || f || ∞ = | f ( t ) | , by compactness. (cid:3) The corollaries below are new results. Before presenting them, adefinition and a theorem appearing in [6] are needed.
Definition 3.2 ([6, Definition 2]) . Let X be a Banach space. Wesay a set F ⊆ X ∗ is a relative boundary if, whenever x ∈ X satisfiessup { f ( x ) : f ∈ F } = 1, there exists f ∈ F such that f ( x ) = 1. Example 3.3.
Any boundary and any w ∗ -compact set will be a relativeboundary. Theorem 3.4 ([6, Theorem 4]) . Let X be a Banach space and supposewe have sets S n ⊆ S X and an increasing sequence H n ⊆ B X ∗ of relativeboundaries, such that S X = S ∞ n =0 S n and the numbers b n = inf { sup { h ( x ) : h ∈ H n } : x ∈ S n } are strictly positive and converge to 1. Then for a suitable sequence ( a n ) ∞ n =0 of numbers the set F = S ∞ n =0 a n ( H n \ H n − ) is a boundary ofan equivalent norm. Given a Banach space with an unconditional basis ( e i ) i ∈ I and x = P i ∈ I x i e i , let e ∗ i ( x ) = x i . For σ ⊆ I , let P σ denote the projection givenby P σ ( x ) = P i ∈ σ e ∗ i ( x ) e i . Corollary 3.5.
Let X have a monotone unconditional basis ( e i ) i ∈ I ,with associated projections P σ , σ ⊆ I , and suppose we can write S X = S ∞ n =1 S n in such a way that the numbers c n = inf { sup {|| P σ ( x ) || : σ ⊆ I, | σ | = n } : x ∈ S n } are strictly positive and converge to 1. Then X admits an equivalent C ∞ smooth norm.Proof. Let H n = { h ∈ B X ∗ : | supp( h ) | ≤ n } . Each H n is a relativeboundary because it is w ∗ -compact. Note that given x ∈ S n and σ ⊆ I , SING BOUNDARIES TO FIND SMOOTH NORMS 13 with | σ | = n , || P σ ( x ) || = sup { f ( P σ ( x )) : f ∈ B X ∗ } = sup { P ∗ σ f ( x ) : f ∈ B X ∗ } . Of course, | supp( P ∗ σ f ) | ≤ n , for all f ∈ B X ∗ . And by monotonicity, || P ∗ σ || = 1. So P ∗ σ ( f ) ∈ H n . Therefore,0 < c n = inf { sup {|| P σ ( x ) || : σ ⊆ I, | σ | = n } : x ∈ S n } = inf { sup { P ∗ σ f ( x ) : f ∈ B X ∗ , σ ⊆ I, | σ | = n } : x ∈ S n } = inf { sup { h ( x ) : h ∈ H n } : x ∈ S n } = b n . Thus, ( b n ) is a strictly positive sequence converging to 1. The set H n \ H n − is w ∗ -LRC, by Example 1.9, (2).By Theorem 3.4, there exists a sequence ( a n ) ∞ n =0 , where the set F = S ∞ n =0 a n ( H n \ H n − ) is a σ - w ∗ -LRC and σ - w ∗ -compact boundary for anequivalent norm ||| · ||| . By Corollary 3.1, X will admit an equivalent C ∞ -smooth that ε -approximates ||| · ||| . (cid:3) Corollary 3.6.
Let X be a Banach space with a monotone uncon-ditional basis ( e i ) i ∈ I and suppose for each x ∈ S X there exists σ ⊂ I, | σ | < ∞ , so that || P σ ( x ) || = 1 . Then X admits an equivalent C ∞ -smooth norm that ε -approximates the original norm.Proof. Let H n = { h ∈ B X ∗ : | supp( h ) | ≤ n } . As mentioned in theproof of Corollary 3.5, each H n is w ∗ -compact and the finite union of w ∗ -LRC sets. Now take x ∈ S X and σ such that || P σ ( x ) || = 1. Thenthere is f ∈ B X ∗ such that1 = || P σ ( x ) || = f ( P σ ( x )) = P ∗ σ f ( x ) . Because ( e i ) i ∈ I is monotone, || P ∗ σ || = 1 and so P ∗ σ f ∈ H | σ | . There-fore, the set H = S ∞ n =0 H n is a boundary satisfying the hypothesis ofCorollary 3.1. (cid:3) Using Corollary 3.5 we can obtain new examples of spaces with equiv-alent C ∞ smooth renormings. Example 3.7.
Let N = S ∞ n =0 A n , where each A n is finite, and let p = ( p n ) be an unbounded increasing sequence of real numbers with p n ≥
1. For each sequence of real numbers x = ( x n ) defineΦ( x ) = sup ( ∞ X n =0 X k ∈ B n | x ( k ) | p n : B n ⊂ A n and B n are pairwise disjoint. ) Proof.
We define ℓ A,p as the space of sequences x where Φ( x/λ ) < ∞ for some λ >
0, with norm || x || = inf { λ > x/λ ) ≤ } . Define thesubspace h A,p as the norm closure of the linear space generated by thebasis e n ( k ) = δ n,k . [6, Example 16] provides an appropriate sequenceof subsets ( S n ) of S X so that Corollary 3.5 holds. (cid:3) Example 3.8.
Let M be an Orlicz function with M ( t ) > t > , and lim t → M ( K ( t ) M ( t ) = + ∞ , for some constant K >
0. Let h M (Γ) be the space of all real functions x defined on Γ with P γ ∈ Γ M ( x γ /ρ ) < ∞ for all ρ >
0, with the norm || x || = inf ( ρ > X γ ∈ Γ M (cid:18) x γ ρ (cid:19) ≤ ) . Proof.
The canonical unit vector basis ( e γ ) γ ∈ Γ of functions e γ ( β ) = δ γ,β is unconditionally monotone. [6, Example 18] provides suitable subsetsof S X to ensure the hypothesis of Corollary 3.5 holds. (cid:3) The final example concerns the predual of a Lorentz sequence space d ( w, , A ), for an arbitrary set A .Let w = ( w n ) ∈ c \ ℓ with each w n strictly positive and w = 1. Wedefine d ( w, , A ) as the space of x : A −→ R for which || x || = sup ( ∞ X j =0 w j | x ( a j ) | : ( a j ) ⊆ A is a sequence of distinct points ) < ∞ . The canonical predual d ∗ ( w, , A ) of d ( w, , A ) is given by the space of y : A −→ R for which y = ( y k ) ∈ c , where y k = sup ( P k − i =0 | y ( a i ) | P k − i =0 w i : a , a , . . . , a k − are distinct points of A ) , with norm || y || = || y || ∞ . We can see that ( e a ) a ∈ A is a monotone uncon-ditional basis for both d ( w, , A ) and d ∗ ( w, , A ). The separable versionof d ∗ ( w, , A ) was first introduced in [13]. SING BOUNDARIES TO FIND SMOOTH NORMS 15
Example 3.9. X = d ∗ ( w, , A ) has a C ∞ smooth equivalent renormingthat ε -approximates the original norm. Proof.
Let y ∈ S X . Since y ∈ c , there exists k ∈ N such that y k = 1 . Itcan also be shown y ∈ c ( A ) and thus the supremum in the definitionof y k is attained. Following this, there exists a , a , . . . , a k − ∈ A suchthat 1 = y k = P k − i =0 | y ( a i ) | P k − i =0 w i . Setting σ = { a , a , ..., a k − } , we have || P σ ( y ) || = 1. By Corollary 3.6, X has a C ∞ smooth equivalent renorming that ε -approximates theoriginal norm. (cid:3) Remark 3.10.
The space X = d ∗ ( w, , A ) for A uncountable is a newexample of a space with a C ∞ smooth renorming. It is not yet knownif X has an analytic renorming. Remark 3.11.
In Theorem 2.1 and Corollary 3.1 we cannot drop the σ - w ∗ -compactness condition in general, and expect an equivalent normof any order of smoothness that depends locally on finitely many co-ordinates. In [4], C ( ω ) is shown to have no such norm. On the otherhand, C ( ω ) admits an equivalent norm supporting a boundary thatis w ∗ -discrete (this follows from [5, Theorem 10]).4. Acknowledgements
The author would like to thank R. J. Smith for discussion and sugges-tions throughout the writing of this paper and S. Troyanski for furtherremarks, in particular bringing his attention to Example 3.9.
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Victor Bible, UCD School of Mathematical Sciences
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