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Smooth partitions of unity on Banach spaces
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Journal of Functional Analysis www.elsevier.com/locate/jfa
Smooth partitions of unity on Banach spaces Michal Johanis 1 Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic
a r t i c l e
i n f o
Article history: Received 20 October 2016 Accepted 24 March 2017 Available online xxxx Communicated by G. Schechtman
a b s t r a c t We show a new characterisation of the existence of smooth partitions of unity on a Banach space. This leads to a slight generalisation of some as well as a very easy recovery of most of the known results using a unified treatment. © 2017 Elsevier Inc. All rights reserved.
MSC: 46B20 46B26 46T20 Keywords: Smooth partitions of unity
Smooth partitions of unity are an important tool in the theory of smooth approximations (see [8, Chapter 7]), smooth extensions, theory of manifolds, and other areas. Clearly a necessary condition for a Banach space to admit smooth partitions of unity is the existence of a smooth bump function. The sufficiency of this condition for a general Banach space is still an open problem. A positive answer was established in many cases, the most important of which are the following (i.e. if one of the conditions below is fulfilled, then the existence of a smooth bump function on X implies that X admits smooth partitions of unity):
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E-mail address: [email protected]ff.cuni.cz. Supported by GAČR 16-07378S.
http://dx.doi.org/10.1016/j.jfa.2017.03.014 0022-1236/© 2017 Elsevier Inc. All rights reserved.
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(i) (ii) (iii) (iv)
X has an SPRI (separable “projectional resolution of the identity”), [6]. X belongs to a P-class, [10]. X = C(K) for K compact, [7]. X has a subspace Y isomorphic to c0 (Γ) such that X/Y admits smooth partitions of unity, [2]. (v) X ∗ is weakly compactly generated (WCG), [12].
For the definition and basic properties of an SPRI see [4, Definition 6.2.6 ff.] or [9, Theorem 3.46]; for the definition of a P-class see p. 6. The original proofs of the results (i), (iv), and (v) use Toruńczyk’s characterisation of the existence of smooth partitions of unity by non-linear homeomorphic embedding into c0 (Γ) with smooth component functions (see e.g. [8, Proposition 7.60]). The other two results use the following theorem of Richard Haydon: Theorem 1 ([10], see also [8, Theorem 7.53]). Let X be a normed linear space that admits a C k -smooth bump function, k ∈ N ∪ {∞}. Let Γ be a set and Φ : X → c0 (Γ) a continuous mapping such that for every γ ∈ Γ the function e∗γ ◦ Φ is C k -smooth. For each finite F ⊂ Γ let PF ∈ C k (X; X) be such that the space span PF (X) admits locally finite C k -partitions of unity. Assume that for each x ∈ X and each ε > 0 there exists δ > 0 such that x − PF (x) < ε if we set F = {γ ∈ Γ; |Φ(x)(γ)| ≥ δ}. Then X admits locally finite and σ-uniformly discrete C k -partitions of unity. While pondering the applicability of Haydon’s theorem we were led to another characterisation of the existence of smooth partitions of unity (Theorem 2). This characterisation allows very easy recovery of all the results above except for the C(K) case. In fact, an immediate consequence is a (at least formal) generalisation of (i), (ii), and (v) given in Corollary 6, which puts all these results under a common roof (this is either obvious or shown in Theorem 10 and Corollary 9). There is also another tiny advantage for the insight into the problem when using Theorem 1: All the original proofs that use Toruńczyk’s characterisation (of course they all come from the same workshop) at some point invoke the completeness of the underlying space, but as we shall see here, the completeness is completely irrelevant to the problem. Before we start, we fix some notation. By U (x, r), resp. B(x, r) we denote the open, resp. closed ball centred at x with radius r. For a function f : X → R we denote suppo f = f −1 (R \ {0}). For other unfamiliar notation or terminology see [8] or [5]. Now, the reason that Haydon’s theorem can be successfully used to prove the wonderful result (iii) is that there is a rich supply of projections of norm one on an Asplund C(K) space (formed by restrictions to clopen subsets of K). So what do we have on an arbitrary Banach space? The projections onto one-dimensional subspaces, of course. This observation leads to the following characterisation:
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Theorem 2. Let X be a normed linear space and k ∈ N ∪ {∞}. The following statements are equivalent: (i) X admits locally finite and σ-uniformly discrete C k -partitions of unity. (ii) X admits a C k -smooth bump and there are a set Γ, a continuous Φ : X → c0 (Γ) such that e∗γ ◦ Φ ∈ C k (X) for every γ ∈ Γ, and vectors {xγn }γ∈Γ,n∈N ⊂ X such that x ∈ span{xγn ; Φ(x)(γ) = 0, n ∈ N} for every x ∈ X. Notice that the condition (ii) resembles a property of a strong Markushevich basis. Before delving into the proof of Theorem 2 we make a short technical intermission. Applications of Theorem 1 involve constructions of continuous mappings into c0 (Γ). To avoid repeating the same argument in several of these constructions we will make use of the following simple lemma. Lemma 3. Let X be a topological space, Γ a set, and Φ : X → RN×Γ . Suppose that all the component functions x → Φ(x)(n, γ) are continuous, limn→∞ Φ(x)(n, γ) = 0 locally uniformly in x ∈ X and uniformly in γ ∈ Γ, and for each fixed n ∈ N, x ∈ X, and ε > 0 there are a neighbourhood U of x and a finite F ⊂ Γ such that |Φ(y)(n, γ)| < ε whenever y ∈ U and γ ∈ Γ \ F . Then Φ is a continuous mapping into c0 (N × Γ). Proof. Fix x ∈ X and ε > 0. There are n0 ∈ N and a neighbourhood U of x such that |Φ(y)(n, γ)| < 2ε whenever n > n0 , y ∈ U , and γ ∈ Γ. For each n ∈ N, n ≤ n0 there are a neighbourhood Vn ⊂ U of x and a finite Fn ⊂ Γ such that |Φ(y)(n, γ)| < 2ε whenever y ∈ Vn and γ ∈ Γ \ Fn . Put F = n≤n0 {n} × Fn and V = n≤n0 Vn . Then F is finite and |Φ(y)(n, γ)| < 2ε whenever y ∈ V and (n, γ) ∈ N × Γ \ F . This shows that Φ maps into c0 (N × Γ). The continuity of Φ follows from the fact that |Φ(y)(n, γ) − Φ(x)(n, γ)| < ε whenever y ∈ V and (n, γ) ∈ N × Γ \ F , and from the continuity of the functions y → Φ(y)(n, γ), (n, γ) ∈ F . 2 Proof of Theorem 2. For the purpose of the proof let us consider the following intermediate statement: (ii)’ X admits a C k -smooth bump and there are a set Λ, a continuous Ψ : X → c0 (Λ) such that e∗λ ◦ Ψ ∈ C k (X) for every λ ∈ Λ, and vectors {xλ }λ∈Λ ⊂ X such that x ∈ {xλ ; Ψ(x)(λ) = 0} for every x ∈ X. (ii)’⇒(i) Since X admits a smooth bump, there are functions hn ∈ C k (X; [0, 1]) such that suppo hn ⊂ U (0, n1 ) and hn (0) > 0. Set Γ = N × Λ and define Φ : X → ∞ (Γ) by
Φ(x)(n, λ) =
1 hn (x − xλ )Ψ(x)(λ). n
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Then Φ is a continuous mapping into c0 (Γ) by Lemma 3. Clearly, e∗(n,λ) ◦ Φ ∈ C k (X) for each (n, λ) ∈ Γ. Next, for each finite non-empty subset F ⊂ Γ let us set m(F ) = max{n ∈ N; (n, λ) ∈ F for some λ ∈ Λ}, let α(F ) ∈ Λ be chosen arbitrarily such that m(F ), α(F ) ∈ F , and let PF : X → X be the linear projection onto span{xα(F ) } of norm at most one. We also set P∅ = 0. We show that the assumptions of Theorem 1 are satisfied. Each one-dimensional subspace of X admits locally finite C k -partitions of 1 unity ([8, Corollary 7.50]). Given x ∈ X and ε > 0 find m ∈ N such that m ≤ 2ε . By the assumption there is α ∈ Λ such that Ψ(x)(α) = 0 and xα is so close to x that hm (x − xα ) > 0. If we set δ = |Φ(x)(m, α)| and F = {(n, λ) ∈ Γ; |Φ(x)(n, λ)| ≥ δ}, then (m, α) ∈ F and hence m(F ) ≥ m. Further, Φ(x) m(F ), α(F ) ≥ δ > 0, and in 1 1 ε particular hm(F ) (x − xα(F ) ) > 0. It follows that x − xα(F ) < m(F ) ≤ m ≤ 2 . Note that PF (xα(F ) ) = xα(F ) and therefore x − PF (x) ≤ x − xα(F ) + PF (xα(F ) ) − PF (x) < ε. (ii)⇒(ii)’ Put Λ = cQ set of all vectors in c00 (Γ × N) with rational 00 (Γ × N), i.e. the coordinates. For each λ ∈ Λ set xλ = γ∈Γ,n∈N λ(γ, n)xγn . Clearly, {xλ ; λ ∈ Λ} = spanQ {xγn ; γ ∈ Γ, n ∈ N}. Further, let q : Q → N be some one-to-one mapping with q(0) = 1 and put m(λ) = max{n ∈ N; λ(γ, n) = 0 for some γ ∈ Γ} for λ ∈ Λ \ {0} and m(0) = 1. Finally, define Ψ : X → RΛ by Ψ(x)(λ) =
m(λ)
1
γ∈Γ,n∈N q λ(γ, n)
Φ(x)(γ).
γ∈Γ : ∃n, λ(γ,n) =0
We claim that Ψ is actually a continuous mapping into c0 (Λ). Indeed, fix x ∈ X and ε > 0. Since Φ is continuous, there are a neighbourhood U of x and a finite set H ⊂ Γ such that Φ(y) < Φ(x) + 1 and |Φ(y)(γ)| < 1 for each y ∈ U and γ ∈ Γ \ H. Note that γ∈Γ : ∃n, λ(γ,n) =0 |Φ(y)(γ)| ≤ (Φ(x) + 1)|H| for any y ∈ U and λ ∈ Λ, and the same holds if we omit any one of the factors in the product. Next, there are a neighbourhood V of x, V ⊂ U , and a finite set E ⊂ Γ such that |Φ(y)(γ)| < ε/(Φ(x) + 1)|H| for each y ∈ V and γ ∈ Γ \ E. Let N ∈ N be such that 1 |H| . Put N < ε/(Φ(x) + 1)
F = λ ∈ Λ; supp λ ⊂ E × {1, . . . , N } and q λ(γ, n) ≤ N for all γ ∈ Γ, n ∈ N and note that F is finite. Now if y ∈ V and λ ∈ Λ \ F , then |Ψ(y)(λ)| < ε. It easily follows that Ψ is a continuous mapping into c0 (Λ). Clearly, e∗λ ◦ Ψ ∈ C k (X) for every λ ∈ Λ. Finally, given x ∈ X and a neighbourhood U of x, by the assumption there is λ ∈ Λ such that xλ ∈ U and Φ(x)(γ) = 0 if λ(γ, n) = 0 for some n ∈ N. Consequently, Ψ(x)(λ) = 0. (i)⇒(ii) The existence of a C k -smooth bump is clear (just take a partition of unity subordinated to a covering of X by U (0, 2) and X \ B(0, 1)). Next, for each n ∈ N let {ϕnλ }λ∈Λ be a locally finite C k -partition of unity on X subordinated to the uniform covering of X by open balls of radius n1 (clearly {ϕnλ } can be constructed by scaling the domains of {ϕ1λ } so that the index set is always the same). Without loss of generality
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we may assume that all the functions ϕnλ are non-zero. We put Γ = N × Λ and define Φ : X → ∞ (Γ) by Φ(x)(n, λ) =
1 ϕnλ (x). n
Then Φ is a continuous mapping into c0 (Γ) by Lemma 3. To finish, choose any xnλ in each suppo ϕnλ . Fix x ∈ X and δ > 0. Let n ∈ N be such that n2 < δ. There is λ ∈ Λ such that x ∈ suppo ϕnλ . Then Φ(x)(n, λ) > 0 and x − xnλ < n2 < δ. It follows that x ∈ {xnλ ; Φ(x)(n, λ) = 0}. 2 As a first application we show how the above characterisation can be used to rather easily obtain the result (iv) from the introduction. Not only that our proof is substantially shorter than the original, but it also does not use any fancy tools like lifting, Bartle–Graves selectors, etc. The stripped-down proof clearly exposes the three main ideas behind it: the use of linear functionals on the subspace Y , so that they can be extended to the whole space; the use of a fundamental biorthogonal system in Y , which allows to link these extensions to functionals on X/Y ; and the crucial property of the norm on c0 (Γ): if we drop all small coordinates, the vector stays close. Corollary 4 ([2]). Let X be a normed linear space and Y ⊂ X a subspace isomorphic to c0 (Γ) for some Γ. If the quotient X/Y admits locally finite C k -partitions of unity for some k ∈ N ∪ {∞}, then X admits locally finite and σ-uniformly discrete C k -partitions of unity. Proof. By extending the equivalent norm from Y we may assume without loss of generality that Y is actually isometric to c0 (Γ). Let Q : X → X/Y be the canonical quotient mapping. Let {(eγ ; fγ )}γ∈Γ be the canonical basis of c0 (Γ) and further assume that each fγ is actually a norm-one functional on X (use the Hahn–Banach theorem). For each n ∈ N let {ψnλ }λ∈Λ be a locally finite C k -partition of unity on X/Y subordinated to the uniform covering of X/Y by open balls of radius n1 (clearly {ψnλ } can be constructed by scaling the domains of {ψ1λ } so that the index set is always the same). Without loss of generality we may assume that all the functions ψnλ are non-zero. Choose znλ ∈ suppo ψnλ and xnλ ∈ X such that Q(xnλ ) = znλ . Let θn ∈ C ∞ R; [0, n1 ] , n ∈ N be Lipschitz functions satisfying θn (t) = 0 if and only if |t| ≤ n2 . We define a mapping Φ : X → ∞ (N × Λ × Γ ∪ N × Λ) by Φ(x)(n, λ, γ) = θn fγ (x − xnλ ) ψnλ Q(x) , 1 Φ(x)(n, λ) = ψnλ Q(x) . n First we show that Φ is actually a continuous mapping into c0 (N × Λ × Γ ∪ N × Λ). Fix x ∈ X and ε > 0. Clearly, 0 ≤ Φ(y)(n, λ, γ) < ε and 0 ≤ Φ(y)(n, λ) < ε for n > 1ε
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and all λ ∈ Λ, γ ∈ Γ, y ∈ X. Now fix n ∈ [1, 1ε ]. Since {ψnλ }λ∈Λ is locally finite, there is a neighbourhood V of Q(x) and a finite F ⊂ Λ such that ψnλ = 0 on V for λ ∈ Λ \ F . Further, there is a neighbourhood U of x such that Q(U ) ⊂ V . Then clearly Φ(y)(n, λ, γ) = Φ(y)(n, λ) = 0 for y ∈ U and λ ∈ Λ \ F , γ ∈ Γ. Now fix λ ∈ F . Assume that ψnλ (Q(x)) = 0. Then Q(x) − znλ < n2 . Put H = {γ ∈ Γ; |fγ (x − xnλ )| ≥ n2 }. We claim that H is finite. Indeed, if H is infinite, then by the w∗ -compactness there is a w∗ -accumulation point f ∈ BX ∗ of {fγ }γ∈H . Then f Y = 0, since {(eγ ; fγ )} is a fundamental biorthogonal system in Y . In particular, f can be considered also as a member of (X/Y )∗ , and so n2 ≤ |f (x − xnλ )| = |f (Q(x) − znλ )| ≤ Q(x) − znλ < n2 , a contradiction. Thus Φ(x)(n, λ, γ) = 0 for γ ∈ Γ \ H. Since the family of functions y → Φ(y)(n, λ, γ), γ ∈ Γ, is equi-continuous, there is a neighbourhood W of x such that |Φ(y)(n, λ, γ)| < ε whenever y ∈ W and γ ∈ Γ \ H. Thus we may apply Lemma 3. Next, we set xnλγ = eγ . Fix any x ∈ X and ε > 0. There is n ∈ N, n ≥ 8ε , and λ ∈ Λ such that ψnλ (Q(x)) > 0 and Q(x) − znλ < 4ε . Thus there is u ∈ Y such that x − xnλ − u < 4ε . Put F = {γ ∈ Γ; |fγ (u)| > 2ε }, which is a finite set (possibly empty), and v = γ∈F fγ (u)eγ . Then u − v ≤ 2ε (we have the supremum norm here) and so x − (xnλ + v) ≤ x − xnλ − u + u − v < ε. Note that |fγ (x − xnλ )| ≥ |fγ (u)| − |fγ (x − xnλ − u)| > 2ε − x − xnλ − u > 4ε ≥ n2 for γ ∈ F . Consequently, Φ(x)(n, λ, γ) > 0 for γ ∈ F . It follows that x ∈ span {xnλ ; Φ(x)(n, λ) = 0} ∪ {xnλγ ; Φ(x)(n, λ, γ) = 0} . Each component of Φ is clearly C k -smooth and the space X admits a C k -smooth bump by [2, Proposition 1]. Thus we may conclude the proof by using Theorem 2. 2
Before going further, we review some notions useful in the study of the (linear) structure of non-separable Banach spaces. Let X be a class of Banach spaces. We say that X is a P-class if for every non-separable X ∈ X there exists a projectional resolution of the identity {Pα }α∈[ω,μ] on X such that (Pα+1 − Pα )(X) ∈ X for all α < μ. We say that X is a P-class if for every non-separable X ∈ X there exists a projectional resolution of the identity {Pα }α∈[ω,μ] on X such that Pα (X) ∈ X for all α < μ. Note that if a class X admits PRI and is closed under complemented subspaces, then X is both P-class and P-class. Therefore reflexive, WCG, WCD, and WLD are all both P-classes and P-classes, as are 1-Plichko spaces ([9, Theorem 5.63]; proof of [11, Theorem 17.6] combined with [11, Theorem 17.16]), spaces with a 1-projectional skeleton (Ondřej Kalenda, private communication; [11, Theorem 17.6]), and duals of Asplund spaces ([3, Remark VI.3.5]). Recall that any space from a P-class has an SPRI ([9, Theorem 3.46]), and any space with an SPRI has a strong Markushevich basis (folklore, see also [9, Theorem 5.1] or the proof of Theorem 8). Although the characterisations of the existence of smooth partitions of unity are inherently non-linear, in all the results from the introduction, except for the C(K) case, the constructions are based on the linear structure in a substantial way. Keeping this in mind, Theorem 2 naturally suggests the following definition:
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Definition 5. Let X be a normed linear space. We say that a system {(xγ ; fγ )}γ∈Γ ⊂ X × X ∗ is a fundamental coordinate system if T (x) = (fγ (x))γ∈Γ is a bounded linear operator from X to c0 (Γ) and x ∈ span{xγ ; fγ (x) = 0} for each x ∈ X. Note that the operator T from the definition is necessarily one-to-one and {fγ }γ∈Γ is bounded (by T ). The following corollary of Theorem 2 is now obvious. Corollary 6. Let X be a normed linear space with a fundamental coordinate system and such that it admits a C k -smooth bump, k ∈ N ∪ {∞}. Then X admits locally finite and σ-uniformly discrete C k -partitions of unity. If X has a strong Markushevich basis, then it also has a fundamental coordinate system (take normalised coordinate functionals). Thus we have an immediate generalisation of the result (i). On the other hand, as we shall see below (Corollary 9), the space JLp , 1 < p < ∞, has a fundamental coordinate system but it does not have a Markushevich basis ([9, Corollary 4.20]). In connection with Corollary 6 and Corollary 4 we remark that the space C(K), where K is a Ciesielski-Pol compact, does not continuously linearly inject into any c0 (Γ) (and so it does not have a fundamental coordinate system), although it has a subspace Y isometric to c0 (Γ1 ) such that the quotient C(K)/Y is isomorphic to c0 (Γ2 ), [3, Theorem VI.8.8.3]. Concerning the result (i) we note that for spaces with an SPRI we do not have to rely on the full construction of a strong Markushevich basis (which is rather hard): Let {Pα }α∈[ω,μ) be an SPRI on X. For each α ∈ [ω, μ) put Qα = Pα+1 − Pα , let {yαn }n∈N be a dense subset of the separable space Qα (X), and let {gαk }k∈N ⊂ Qα (X)∗ be separating for Qα (X). Put fαk = gαk ◦ Qα /(gαk ◦ Qα + 1), Γ = [ω, μ) × N × N, and 1 define T : X → ∞ (Γ) by T (x)(α, k, n) = kn fαk (x). Then T is clearly a bounded linear operator. Further, set xγ = yαn for γ = (α, k, n) ∈ Γ. Fix x ∈ X. Since Qα (x) = 0 if and only if there is k ∈ N such that gαk (Qα (x)) = 0, which is equivalent to fαk (x) = 0, we have x ∈ span{Qα (x); α ∈ [ω, μ)} = span{Qα (x); α ∈ [ω, μ), Qα (x) = 0}
⊂ span {yαn }n∈N ; α ∈ [ω, μ), ∃k ∈ N : fαk (x) = 0 = span{yαn ; n ∈ N, α ∈ [ω, μ), ∃k ∈ N : T (x)(α, k, n) = 0} = span{xγ ; T (x)(γ) = 0}. Note that Qβ ◦ Qα = 0 for β = α. Hence, given α ∈ [ω, μ) and n ∈ N, we 1 1 1 have T (yαn )(β, k, m) = km fβk (yαn ) = km fβk (Qα (yαn )) = km gβk Qβ Qα (yαn ) / 1 1 |fαk (yαn )| ≤ km yαn . (gβk ◦ Qβ + 1) = 0 for β = α. Also, |T (yαn )(α, k, m)| = km Thus T (yαn ) ∈ c0 (Γ). Since we have seen above that X = span{yαn ; α ∈ [ω, μ), n ∈ N}, it follows that T maps into c0 (Γ) and so X has a fundamental coordinate system.
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We proceed by deducing the result (v) from Corollary 9 and the result (ii) from Theorem 10. We start with an easy observation. Fact 7. Let X be a normed linear space and {(xγ ; fγ )}γ∈Γ ⊂ X × X ∗ . Then x ∈ ∗ span{xγ ; fγ (x) = 0} for every x ∈ X if and only if f ∈ spanw {fγ ; f (xγ ) = 0} for every f ∈ X ∗ . Proof. ⇒ Assume that it is not true for some f ∈ X ∗ . Denote A = {γ ∈ Γ; f (xγ ) = 0}. By the separation theorem there is x ∈ X such that f (x) = 0 and fγ (x) = 0 for each γ ∈ A. It follows that x ∈ span{xγ ; fγ (x) = 0} ⊂ span{xγ ; γ ∈ Γ \ A} ⊂ {f }⊥ , a contradiction. ⇐ Assume that it is not true for some x ∈ X. Denote A = {γ ∈ Γ; fγ (x) = 0}. By the separation theorem there is f ∈ X ∗ such that f (x) = 0 and f (xγ ) = 0 for each ∗ ∗ γ ∈ A. It follows that f ∈ spanw {fγ ; f (xγ ) = 0} ⊂ spanw {fγ ; γ ∈ Γ \ A} ⊂ {x}⊥ , a contradiction. 2 The first part of the next theorem is probably folklore among experts. We include the proof for the convenience of the reader. Theorem 8. Let X be a WCG Banach space and let K ⊂ X be a weakly compact convex symmetric set that generates the space X. Then X has a strong Markushevich basis {(xγ ; fγ )}γ∈Γ ⊂ K × X ∗ . Such a basis has the following properties: T (f ) = (f (xγ ))γ∈Γ ∗ is a bounded linear operator from X ∗ to c0 (Γ) and f ∈ spanw {fγ ; f (xγ ) = 0} for each f ∈ X ∗. Proof. We prove the first part by transfinite induction on dens X. Suppose first that X is separable. Let {zn }n∈N ⊂ K be a dense set in K and {hn }n∈N a norming set in X ∗ . Note that span{zn } = X. By [5, Theorem 4.59] there is a Markushevich basis {(yn ; gn )}n∈N of X such that span{yn } = span{zn } and span{gn } = span{hn }. In particular, this basis is norming. Hence by [9, Theorem 1.42] there is a strong Markushevich basis {(xn ; fn )}n∈N of X such that {xn } ⊂ span{yn } = span{zn }. Since span{zn } ⊂ n∈N nK, by scaling we may assume that {xn } ⊂ K. Now assume that dens X > ω and the statement is true for all WCG spaces of density less than dens X. By [5, Theorem 13.6] there is a PRI {Pα }α∈[ω,μ] on X such that Pα (K) ⊂ K for each α ∈ [ω, μ]. Denote Qα = Pα+1 − Pα . For each α ∈ [ω, μ) the space Qα (X) is of density at most card α < dens X and is generated by the weakly compact convex symmetric set 12 Qα (K). Thus by the inductive hypothesis Qα (X) has 1 α α a strong Markushevich basis {(xα γ ; gγ )}γ∈Γα such that {xγ }γ∈Γα ⊂ 2 Qα (K) ⊂ K. Put α α α α fγ = gγ ◦ Qα . We claim that {(xγ ; fγ )}α∈[ω,μ),γ∈Γα is a strong Markushevich basis of X. Indeed, Qα (xβη ) = Qα (Qβ (xβη )) = 0 and hence fγα (xβη ) = gγα (Qα (xβη )) = 0 for α = β. α α α α Further, fγα (xα η ) = gγ (Qα (xη )) = gγ (xη ) = δγ,η (the Kronecker delta). Fix any x ∈ X. α α Then Qα (x) ∈ span{xγ ; γ ∈ Γα : gγα (Qα (x)) = 0} = span{xα γ ; γ ∈ Γα : fγ (x) = 0}.
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α Hence x ∈ span{Qα (x); α ∈ [ω, μ)} ⊂ span α∈[ω,μ) span{xα γ ; γ ∈ Γα : fγ (x) = 0} ⊂ α span{xα γ ; α ∈ [ω, μ), γ ∈ Γα : fγ (x) = 0}. Finally, note that this strongness property implies that the biorthogonal system is total. To prove the second part of the theorem, denote by τ the topology on X ∗ given by the uniform convergence on K. Put T (f ) = (f (xγ ))γ∈Γ for f ∈ X ∗ . Then T is clearly a bounded linear operator from X ∗ to ∞ (Γ). Since T (f ) = supγ∈Γ |f (xγ )| ≤ supx∈K |f (x)|, the operator T is moreover τ - · continuous. Further, T (fα ) ∈ c00 (Γ) ∗ for every α ∈ Γ. By the Mackey–Arens theorem, spanτ {fα } = spanw {fα } = X ∗ . Consequently, T (X ∗ ) = T (spanτ {fα }) ⊂ span{T (fα )} ⊂ c0 (Γ). The rest follows from Fact 7. 2 We remark that the heart of the construction of a strong Markushevich basis lies in the separable case and is seriously difficult. The strongness of the PRI then arranges the rest. However, for our purpose (the second part of the previous theorem), the full strongness (and even the biorthogonality) of the Markushevich basis is not necessary. It would be sufficient to carry the required properties through the transfinite induction and use just the strongness provided by the PRI. The weak compactness is indispensable though. Corollary 9. Let X be a normed linear space such that X ∗ is WCG. Then X has a fundamental coordinate system. Proof. Let {(fγ ; Fγ )}γ∈Γ ⊂ X ∗ × X ∗∗ be a Markushevich basis from Theorem 8. Note w∗
that {fγ }γ∈Γ is bounded. Fix γ ∈ Γ. Then by the Goldstine theorem Fγ ∈ B for ∗ some ball B ⊂ X. Since X has the property C ([5, Definition 14.32, Theorem 14.31]), by [5, Theorem 14.37] there is a countable set of vectors {xγn }n∈N ⊂ B such that ∗ Fγ ∈ convw {xγn }n∈N . We claim that {(xγn ; n1 fγ )}γ∈Γ,n∈N is a fundamental coordinate system. Indeed, T (x) = n1 fγ (x) γ∈Γ,n∈N is a bounded linear operator from X to c0 (Γ × N), as (fγ (x))γ∈Γ ∈ c0 (Γ) by Theorem 8. Fix x ∈ X and denote A = {xγn ; fγ (x) = 0, n ∈ N}. ∗ Theorem 8 implies that F ∈ spanw {Fγ ; F (fγ ) = 0} for any F ∈ X ∗∗ and so ∗
x ∈ spanw {Fγ ; fγ (x) = 0} ⊂ spanw
∗
∗
∗
convw {xγn }n∈N = spanw A.
γ∈Γ : fγ (x) =0
But since x ∈ X and span A ⊂ X, this means that x ∈ spanw A = spanA. 2 We note that there is a Banach space X such that it is a second dual space, it has an equivalent C 1 -smooth norm, X ∗ is a subspace of a Hilbert-generated space (in particular a subspace of a WCG space), and there is no bounded linear one-to-one operator from X to c0 (Γ), [1]. Therefore there is no hope for generalising the result (v) beyond the dual being WCG using the approach above (or the original proof as well – both result in a linear injection into c0 (Γ)).
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Theorem 10. Every Banach space that belongs to a P-class has a fundamental coordinate system. Proof. Let X be a P-class and X ∈ X . We use transfinite induction on dens X. If X is separable, then we can use the existence of a strong Markushevich basis. However, this difficult result is not necessary. A direct construction is as follows: Let {yn }n∈N ⊂ X be dense in X and let {gn }n∈N ⊂ X ∗ be such that it separates the points of X and gn ≤ n1 . For k, n ∈ N put xkn = yn and fkn = n1 gk . Then {(xkn , fkn )}k,n∈N is a fundamental 1 coordinate system: Fix x ∈ X. Then |fkn (x)| ≤ nk x. Also, there is m ∈ N such that gm (x) = 0 and so x ∈ span{yn ; n ∈ N} = span{xmn ; n ∈ N} ⊂ span{xkn ; fkn (x) = 0}. Now assume that dens X > ω and every space in X of density less than dens X has a fundamental coordinate system. Let {Pα }α∈[ω,μ] be a PRI on X such that Pα (X) ∈ X for α ∈ [ω, μ). Put Qα = Pα+1 − Pα . By the inductive hypothesis, for each α ∈ [ω, μ) there α is a fundamental coordinate system {(xα γ ; gγ )}γ∈Γα on Pα (X) and there is Kα > 0 such 1 α that {gγ }γ∈Γα ⊂ B(0, Kα ). Since Qα (X) ⊂ Pα+1 (X), we may set fγα+1 = Kα+1 gγα+1 ◦Qα and note that fγα+1 ≤ 2. We claim that {(xα+1 ; fγα+1 )}α∈[ω,μ),γ∈Γα+1 is a fundamental γ coordinate system on X. Indeed, the formula T (x) = (fγα+1 (x))α∈[ω,μ),γ∈Γα+1 clearly defines a bounded lin ear operator from X to ∞ (Γ), where Γ = α∈[ω,μ) {α} × Γα+1 . Now fix x ∈ X and ε > 0. Then the set A = {α ∈ [ω, μ); Qα (x) > ε} is finite. So, |fγα+1 (x)| ≤ 1 α+1 Qα (x) ≤ ε whenever α ∈ [ω, μ) \ A and γ ∈ Γα+1 . On the other hand, if Kα+1 gγ
α ∈ A, then the set {γ ∈ Γα+1 ; |fγα+1 (x)| > ε} = γ ∈ Γα+1 ; |gγα+1 (Qα (x))| > Kα+1 ε is finite by the definition of a fundamental coordinate system. Finally, as Qα (x) ∈ Pα+1 (X), the assumption gives Qα (x) ∈ span{xα+1 ; gγα+1 (Qα (x)) = 0} = span{xα+1 ; fγα+1 (x) = 0}. γ γ α+1 α+1 Therefore x ∈ span{Qα (x); α ∈ [ω, μ)} ⊂ span α∈[ω,μ) span{xγ ; fγ (x) = 0} ⊂ span{xα+1 ; fγα+1 (x) = 0}. 2 γ References [1] Spiros A. Argyros, Sophocles Mercourakis, A note on the structure of WUR Banach spaces, Comment. Math. Univ. Carolin. 46 (3) (2005) 399–408. [2] Robert Deville, Gilles Godefroy, Václav Zizler, The three space problem for smooth partitions of unity and C(K) spaces, Math. Ann. 288 (1) (1990) 613–625. [3] Robert Deville, Gilles Godefroy, Václav Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surv. Pure Appl. Math., vol. 64, Longman Scientific & Technical, Harlow, 1993. [4] Marián Fabian, Gâteaux Differentiability of Convex Functions and Topology – Weak Asplund Spaces, CMS Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1997. [5] Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books Math., Springer, New York, 2011. [6] Gilles Godefroy, Stanimir L. Troyanski, John H.M. Whitfield, Václav Zizler, Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (3) (1983) 344–352. [7] Petr Hájek, Richard Haydon, Smooth norms and approximation in Banach spaces of the type C(K), Q. J. Math. 58 (2) (2007) 221–228. [8] Petr Hájek, Michal Johanis, Smooth Analysis in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 19, Walter de Gruyter, Berlin, 2014. [9] Petr Hájek, Vicente Montesinos, Jon Vanderwerff, Václav Zizler, Biorthogonal Systems in Banach Spaces, CMS Books Math., vol. 26, Springer, New York, 2008.
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[10] Richard Haydon, Smooth functions and partitions of unity on certain Banach spaces, Q. J. Math. 47 (4) (1996) 455–468. [11] Jerzy Kąkol, Wiesław Kubiś, Manuel López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Dev. Math., vol. 24, Springer, New York, 2011. [12] David P. McLaughlin, Smooth partitions of unity in preduals of WCG spaces, Math. Z. 211 (1992) 189–194.
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Contents lists available at ScienceDirect
Journal of Functional Analysis www.elsevier.com/locate/jfa
Smooth partitions of unity on Banach spaces Michal Johanis 1 Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Praha 8, Czech Republic
a r t i c l e
i n f o
Article history: Received 20 October 2016 Accepted 24 March 2017 Available online xxxx Communicated by G. Schechtman
a b s t r a c t We show a new characterisation of the existence of smooth partitions of unity on a Banach space. This leads to a slight generalisation of some as well as a very easy recovery of most of the known results using a unified treatment. © 2017 Elsevier Inc. All rights reserved.
MSC: 46B20 46B26 46T20 Keywords: Smooth partitions of unity
Smooth partitions of unity are an important tool in the theory of smooth approximations (see [8, Chapter 7]), smooth extensions, theory of manifolds, and other areas. Clearly a necessary condition for a Banach space to admit smooth partitions of unity is the existence of a smooth bump function. The sufficiency of this condition for a general Banach space is still an open problem. A positive answer was established in many cases, the most important of which are the following (i.e. if one of the conditions below is fulfilled, then the existence of a smooth bump function on X implies that X admits smooth partitions of unity):
1
E-mail address: [email protected]ff.cuni.cz. Supported by GAČR 16-07378S.
http://dx.doi.org/10.1016/j.jfa.2017.03.014 0022-1236/© 2017 Elsevier Inc. All rights reserved.
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(i) (ii) (iii) (iv)
X has an SPRI (separable “projectional resolution of the identity”), [6]. X belongs to a P-class, [10]. X = C(K) for K compact, [7]. X has a subspace Y isomorphic to c0 (Γ) such that X/Y admits smooth partitions of unity, [2]. (v) X ∗ is weakly compactly generated (WCG), [12].
For the definition and basic properties of an SPRI see [4, Definition 6.2.6 ff.] or [9, Theorem 3.46]; for the definition of a P-class see p. 6. The original proofs of the results (i), (iv), and (v) use Toruńczyk’s characterisation of the existence of smooth partitions of unity by non-linear homeomorphic embedding into c0 (Γ) with smooth component functions (see e.g. [8, Proposition 7.60]). The other two results use the following theorem of Richard Haydon: Theorem 1 ([10], see also [8, Theorem 7.53]). Let X be a normed linear space that admits a C k -smooth bump function, k ∈ N ∪ {∞}. Let Γ be a set and Φ : X → c0 (Γ) a continuous mapping such that for every γ ∈ Γ the function e∗γ ◦ Φ is C k -smooth. For each finite F ⊂ Γ let PF ∈ C k (X; X) be such that the space span PF (X) admits locally finite C k -partitions of unity. Assume that for each x ∈ X and each ε > 0 there exists δ > 0 such that x − PF (x) < ε if we set F = {γ ∈ Γ; |Φ(x)(γ)| ≥ δ}. Then X admits locally finite and σ-uniformly discrete C k -partitions of unity. While pondering the applicability of Haydon’s theorem we were led to another characterisation of the existence of smooth partitions of unity (Theorem 2). This characterisation allows very easy recovery of all the results above except for the C(K) case. In fact, an immediate consequence is a (at least formal) generalisation of (i), (ii), and (v) given in Corollary 6, which puts all these results under a common roof (this is either obvious or shown in Theorem 10 and Corollary 9). There is also another tiny advantage for the insight into the problem when using Theorem 1: All the original proofs that use Toruńczyk’s characterisation (of course they all come from the same workshop) at some point invoke the completeness of the underlying space, but as we shall see here, the completeness is completely irrelevant to the problem. Before we start, we fix some notation. By U (x, r), resp. B(x, r) we denote the open, resp. closed ball centred at x with radius r. For a function f : X → R we denote suppo f = f −1 (R \ {0}). For other unfamiliar notation or terminology see [8] or [5]. Now, the reason that Haydon’s theorem can be successfully used to prove the wonderful result (iii) is that there is a rich supply of projections of norm one on an Asplund C(K) space (formed by restrictions to clopen subsets of K). So what do we have on an arbitrary Banach space? The projections onto one-dimensional subspaces, of course. This observation leads to the following characterisation:
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Theorem 2. Let X be a normed linear space and k ∈ N ∪ {∞}. The following statements are equivalent: (i) X admits locally finite and σ-uniformly discrete C k -partitions of unity. (ii) X admits a C k -smooth bump and there are a set Γ, a continuous Φ : X → c0 (Γ) such that e∗γ ◦ Φ ∈ C k (X) for every γ ∈ Γ, and vectors {xγn }γ∈Γ,n∈N ⊂ X such that x ∈ span{xγn ; Φ(x)(γ) = 0, n ∈ N} for every x ∈ X. Notice that the condition (ii) resembles a property of a strong Markushevich basis. Before delving into the proof of Theorem 2 we make a short technical intermission. Applications of Theorem 1 involve constructions of continuous mappings into c0 (Γ). To avoid repeating the same argument in several of these constructions we will make use of the following simple lemma. Lemma 3. Let X be a topological space, Γ a set, and Φ : X → RN×Γ . Suppose that all the component functions x → Φ(x)(n, γ) are continuous, limn→∞ Φ(x)(n, γ) = 0 locally uniformly in x ∈ X and uniformly in γ ∈ Γ, and for each fixed n ∈ N, x ∈ X, and ε > 0 there are a neighbourhood U of x and a finite F ⊂ Γ such that |Φ(y)(n, γ)| < ε whenever y ∈ U and γ ∈ Γ \ F . Then Φ is a continuous mapping into c0 (N × Γ). Proof. Fix x ∈ X and ε > 0. There are n0 ∈ N and a neighbourhood U of x such that |Φ(y)(n, γ)| < 2ε whenever n > n0 , y ∈ U , and γ ∈ Γ. For each n ∈ N, n ≤ n0 there are a neighbourhood Vn ⊂ U of x and a finite Fn ⊂ Γ such that |Φ(y)(n, γ)| < 2ε whenever y ∈ Vn and γ ∈ Γ \ Fn . Put F = n≤n0 {n} × Fn and V = n≤n0 Vn . Then F is finite and |Φ(y)(n, γ)| < 2ε whenever y ∈ V and (n, γ) ∈ N × Γ \ F . This shows that Φ maps into c0 (N × Γ). The continuity of Φ follows from the fact that |Φ(y)(n, γ) − Φ(x)(n, γ)| < ε whenever y ∈ V and (n, γ) ∈ N × Γ \ F , and from the continuity of the functions y → Φ(y)(n, γ), (n, γ) ∈ F . 2 Proof of Theorem 2. For the purpose of the proof let us consider the following intermediate statement: (ii)’ X admits a C k -smooth bump and there are a set Λ, a continuous Ψ : X → c0 (Λ) such that e∗λ ◦ Ψ ∈ C k (X) for every λ ∈ Λ, and vectors {xλ }λ∈Λ ⊂ X such that x ∈ {xλ ; Ψ(x)(λ) = 0} for every x ∈ X. (ii)’⇒(i) Since X admits a smooth bump, there are functions hn ∈ C k (X; [0, 1]) such that suppo hn ⊂ U (0, n1 ) and hn (0) > 0. Set Γ = N × Λ and define Φ : X → ∞ (Γ) by
Φ(x)(n, λ) =
1 hn (x − xλ )Ψ(x)(λ). n
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Then Φ is a continuous mapping into c0 (Γ) by Lemma 3. Clearly, e∗(n,λ) ◦ Φ ∈ C k (X) for each (n, λ) ∈ Γ. Next, for each finite non-empty subset F ⊂ Γ let us set m(F ) = max{n ∈ N; (n, λ) ∈ F for some λ ∈ Λ}, let α(F ) ∈ Λ be chosen arbitrarily such that m(F ), α(F ) ∈ F , and let PF : X → X be the linear projection onto span{xα(F ) } of norm at most one. We also set P∅ = 0. We show that the assumptions of Theorem 1 are satisfied. Each one-dimensional subspace of X admits locally finite C k -partitions of 1 unity ([8, Corollary 7.50]). Given x ∈ X and ε > 0 find m ∈ N such that m ≤ 2ε . By the assumption there is α ∈ Λ such that Ψ(x)(α) = 0 and xα is so close to x that hm (x − xα ) > 0. If we set δ = |Φ(x)(m, α)| and F = {(n, λ) ∈ Γ; |Φ(x)(n, λ)| ≥ δ}, then (m, α) ∈ F and hence m(F ) ≥ m. Further, Φ(x) m(F ), α(F ) ≥ δ > 0, and in 1 1 ε particular hm(F ) (x − xα(F ) ) > 0. It follows that x − xα(F ) < m(F ) ≤ m ≤ 2 . Note that PF (xα(F ) ) = xα(F ) and therefore x − PF (x) ≤ x − xα(F ) + PF (xα(F ) ) − PF (x) < ε. (ii)⇒(ii)’ Put Λ = cQ set of all vectors in c00 (Γ × N) with rational 00 (Γ × N), i.e. the coordinates. For each λ ∈ Λ set xλ = γ∈Γ,n∈N λ(γ, n)xγn . Clearly, {xλ ; λ ∈ Λ} = spanQ {xγn ; γ ∈ Γ, n ∈ N}. Further, let q : Q → N be some one-to-one mapping with q(0) = 1 and put m(λ) = max{n ∈ N; λ(γ, n) = 0 for some γ ∈ Γ} for λ ∈ Λ \ {0} and m(0) = 1. Finally, define Ψ : X → RΛ by Ψ(x)(λ) =
m(λ)
1
γ∈Γ,n∈N q λ(γ, n)
Φ(x)(γ).
γ∈Γ : ∃n, λ(γ,n) =0
We claim that Ψ is actually a continuous mapping into c0 (Λ). Indeed, fix x ∈ X and ε > 0. Since Φ is continuous, there are a neighbourhood U of x and a finite set H ⊂ Γ such that Φ(y) < Φ(x) + 1 and |Φ(y)(γ)| < 1 for each y ∈ U and γ ∈ Γ \ H. Note that γ∈Γ : ∃n, λ(γ,n) =0 |Φ(y)(γ)| ≤ (Φ(x) + 1)|H| for any y ∈ U and λ ∈ Λ, and the same holds if we omit any one of the factors in the product. Next, there are a neighbourhood V of x, V ⊂ U , and a finite set E ⊂ Γ such that |Φ(y)(γ)| < ε/(Φ(x) + 1)|H| for each y ∈ V and γ ∈ Γ \ E. Let N ∈ N be such that 1 |H| . Put N < ε/(Φ(x) + 1)
F = λ ∈ Λ; supp λ ⊂ E × {1, . . . , N } and q λ(γ, n) ≤ N for all γ ∈ Γ, n ∈ N and note that F is finite. Now if y ∈ V and λ ∈ Λ \ F , then |Ψ(y)(λ)| < ε. It easily follows that Ψ is a continuous mapping into c0 (Λ). Clearly, e∗λ ◦ Ψ ∈ C k (X) for every λ ∈ Λ. Finally, given x ∈ X and a neighbourhood U of x, by the assumption there is λ ∈ Λ such that xλ ∈ U and Φ(x)(γ) = 0 if λ(γ, n) = 0 for some n ∈ N. Consequently, Ψ(x)(λ) = 0. (i)⇒(ii) The existence of a C k -smooth bump is clear (just take a partition of unity subordinated to a covering of X by U (0, 2) and X \ B(0, 1)). Next, for each n ∈ N let {ϕnλ }λ∈Λ be a locally finite C k -partition of unity on X subordinated to the uniform covering of X by open balls of radius n1 (clearly {ϕnλ } can be constructed by scaling the domains of {ϕ1λ } so that the index set is always the same). Without loss of generality
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we may assume that all the functions ϕnλ are non-zero. We put Γ = N × Λ and define Φ : X → ∞ (Γ) by Φ(x)(n, λ) =
1 ϕnλ (x). n
Then Φ is a continuous mapping into c0 (Γ) by Lemma 3. To finish, choose any xnλ in each suppo ϕnλ . Fix x ∈ X and δ > 0. Let n ∈ N be such that n2 < δ. There is λ ∈ Λ such that x ∈ suppo ϕnλ . Then Φ(x)(n, λ) > 0 and x − xnλ < n2 < δ. It follows that x ∈ {xnλ ; Φ(x)(n, λ) = 0}. 2 As a first application we show how the above characterisation can be used to rather easily obtain the result (iv) from the introduction. Not only that our proof is substantially shorter than the original, but it also does not use any fancy tools like lifting, Bartle–Graves selectors, etc. The stripped-down proof clearly exposes the three main ideas behind it: the use of linear functionals on the subspace Y , so that they can be extended to the whole space; the use of a fundamental biorthogonal system in Y , which allows to link these extensions to functionals on X/Y ; and the crucial property of the norm on c0 (Γ): if we drop all small coordinates, the vector stays close. Corollary 4 ([2]). Let X be a normed linear space and Y ⊂ X a subspace isomorphic to c0 (Γ) for some Γ. If the quotient X/Y admits locally finite C k -partitions of unity for some k ∈ N ∪ {∞}, then X admits locally finite and σ-uniformly discrete C k -partitions of unity. Proof. By extending the equivalent norm from Y we may assume without loss of generality that Y is actually isometric to c0 (Γ). Let Q : X → X/Y be the canonical quotient mapping. Let {(eγ ; fγ )}γ∈Γ be the canonical basis of c0 (Γ) and further assume that each fγ is actually a norm-one functional on X (use the Hahn–Banach theorem). For each n ∈ N let {ψnλ }λ∈Λ be a locally finite C k -partition of unity on X/Y subordinated to the uniform covering of X/Y by open balls of radius n1 (clearly {ψnλ } can be constructed by scaling the domains of {ψ1λ } so that the index set is always the same). Without loss of generality we may assume that all the functions ψnλ are non-zero. Choose znλ ∈ suppo ψnλ and xnλ ∈ X such that Q(xnλ ) = znλ . Let θn ∈ C ∞ R; [0, n1 ] , n ∈ N be Lipschitz functions satisfying θn (t) = 0 if and only if |t| ≤ n2 . We define a mapping Φ : X → ∞ (N × Λ × Γ ∪ N × Λ) by Φ(x)(n, λ, γ) = θn fγ (x − xnλ ) ψnλ Q(x) , 1 Φ(x)(n, λ) = ψnλ Q(x) . n First we show that Φ is actually a continuous mapping into c0 (N × Λ × Γ ∪ N × Λ). Fix x ∈ X and ε > 0. Clearly, 0 ≤ Φ(y)(n, λ, γ) < ε and 0 ≤ Φ(y)(n, λ) < ε for n > 1ε
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and all λ ∈ Λ, γ ∈ Γ, y ∈ X. Now fix n ∈ [1, 1ε ]. Since {ψnλ }λ∈Λ is locally finite, there is a neighbourhood V of Q(x) and a finite F ⊂ Λ such that ψnλ = 0 on V for λ ∈ Λ \ F . Further, there is a neighbourhood U of x such that Q(U ) ⊂ V . Then clearly Φ(y)(n, λ, γ) = Φ(y)(n, λ) = 0 for y ∈ U and λ ∈ Λ \ F , γ ∈ Γ. Now fix λ ∈ F . Assume that ψnλ (Q(x)) = 0. Then Q(x) − znλ < n2 . Put H = {γ ∈ Γ; |fγ (x − xnλ )| ≥ n2 }. We claim that H is finite. Indeed, if H is infinite, then by the w∗ -compactness there is a w∗ -accumulation point f ∈ BX ∗ of {fγ }γ∈H . Then f Y = 0, since {(eγ ; fγ )} is a fundamental biorthogonal system in Y . In particular, f can be considered also as a member of (X/Y )∗ , and so n2 ≤ |f (x − xnλ )| = |f (Q(x) − znλ )| ≤ Q(x) − znλ < n2 , a contradiction. Thus Φ(x)(n, λ, γ) = 0 for γ ∈ Γ \ H. Since the family of functions y → Φ(y)(n, λ, γ), γ ∈ Γ, is equi-continuous, there is a neighbourhood W of x such that |Φ(y)(n, λ, γ)| < ε whenever y ∈ W and γ ∈ Γ \ H. Thus we may apply Lemma 3. Next, we set xnλγ = eγ . Fix any x ∈ X and ε > 0. There is n ∈ N, n ≥ 8ε , and λ ∈ Λ such that ψnλ (Q(x)) > 0 and Q(x) − znλ < 4ε . Thus there is u ∈ Y such that x − xnλ − u < 4ε . Put F = {γ ∈ Γ; |fγ (u)| > 2ε }, which is a finite set (possibly empty), and v = γ∈F fγ (u)eγ . Then u − v ≤ 2ε (we have the supremum norm here) and so x − (xnλ + v) ≤ x − xnλ − u + u − v < ε. Note that |fγ (x − xnλ )| ≥ |fγ (u)| − |fγ (x − xnλ − u)| > 2ε − x − xnλ − u > 4ε ≥ n2 for γ ∈ F . Consequently, Φ(x)(n, λ, γ) > 0 for γ ∈ F . It follows that x ∈ span {xnλ ; Φ(x)(n, λ) = 0} ∪ {xnλγ ; Φ(x)(n, λ, γ) = 0} . Each component of Φ is clearly C k -smooth and the space X admits a C k -smooth bump by [2, Proposition 1]. Thus we may conclude the proof by using Theorem 2. 2
Before going further, we review some notions useful in the study of the (linear) structure of non-separable Banach spaces. Let X be a class of Banach spaces. We say that X is a P-class if for every non-separable X ∈ X there exists a projectional resolution of the identity {Pα }α∈[ω,μ] on X such that (Pα+1 − Pα )(X) ∈ X for all α < μ. We say that X is a P-class if for every non-separable X ∈ X there exists a projectional resolution of the identity {Pα }α∈[ω,μ] on X such that Pα (X) ∈ X for all α < μ. Note that if a class X admits PRI and is closed under complemented subspaces, then X is both P-class and P-class. Therefore reflexive, WCG, WCD, and WLD are all both P-classes and P-classes, as are 1-Plichko spaces ([9, Theorem 5.63]; proof of [11, Theorem 17.6] combined with [11, Theorem 17.16]), spaces with a 1-projectional skeleton (Ondřej Kalenda, private communication; [11, Theorem 17.6]), and duals of Asplund spaces ([3, Remark VI.3.5]). Recall that any space from a P-class has an SPRI ([9, Theorem 3.46]), and any space with an SPRI has a strong Markushevich basis (folklore, see also [9, Theorem 5.1] or the proof of Theorem 8). Although the characterisations of the existence of smooth partitions of unity are inherently non-linear, in all the results from the introduction, except for the C(K) case, the constructions are based on the linear structure in a substantial way. Keeping this in mind, Theorem 2 naturally suggests the following definition:
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Definition 5. Let X be a normed linear space. We say that a system {(xγ ; fγ )}γ∈Γ ⊂ X × X ∗ is a fundamental coordinate system if T (x) = (fγ (x))γ∈Γ is a bounded linear operator from X to c0 (Γ) and x ∈ span{xγ ; fγ (x) = 0} for each x ∈ X. Note that the operator T from the definition is necessarily one-to-one and {fγ }γ∈Γ is bounded (by T ). The following corollary of Theorem 2 is now obvious. Corollary 6. Let X be a normed linear space with a fundamental coordinate system and such that it admits a C k -smooth bump, k ∈ N ∪ {∞}. Then X admits locally finite and σ-uniformly discrete C k -partitions of unity. If X has a strong Markushevich basis, then it also has a fundamental coordinate system (take normalised coordinate functionals). Thus we have an immediate generalisation of the result (i). On the other hand, as we shall see below (Corollary 9), the space JLp , 1 < p < ∞, has a fundamental coordinate system but it does not have a Markushevich basis ([9, Corollary 4.20]). In connection with Corollary 6 and Corollary 4 we remark that the space C(K), where K is a Ciesielski-Pol compact, does not continuously linearly inject into any c0 (Γ) (and so it does not have a fundamental coordinate system), although it has a subspace Y isometric to c0 (Γ1 ) such that the quotient C(K)/Y is isomorphic to c0 (Γ2 ), [3, Theorem VI.8.8.3]. Concerning the result (i) we note that for spaces with an SPRI we do not have to rely on the full construction of a strong Markushevich basis (which is rather hard): Let {Pα }α∈[ω,μ) be an SPRI on X. For each α ∈ [ω, μ) put Qα = Pα+1 − Pα , let {yαn }n∈N be a dense subset of the separable space Qα (X), and let {gαk }k∈N ⊂ Qα (X)∗ be separating for Qα (X). Put fαk = gαk ◦ Qα /(gαk ◦ Qα + 1), Γ = [ω, μ) × N × N, and 1 define T : X → ∞ (Γ) by T (x)(α, k, n) = kn fαk (x). Then T is clearly a bounded linear operator. Further, set xγ = yαn for γ = (α, k, n) ∈ Γ. Fix x ∈ X. Since Qα (x) = 0 if and only if there is k ∈ N such that gαk (Qα (x)) = 0, which is equivalent to fαk (x) = 0, we have x ∈ span{Qα (x); α ∈ [ω, μ)} = span{Qα (x); α ∈ [ω, μ), Qα (x) = 0}
⊂ span {yαn }n∈N ; α ∈ [ω, μ), ∃k ∈ N : fαk (x) = 0 = span{yαn ; n ∈ N, α ∈ [ω, μ), ∃k ∈ N : T (x)(α, k, n) = 0} = span{xγ ; T (x)(γ) = 0}. Note that Qβ ◦ Qα = 0 for β = α. Hence, given α ∈ [ω, μ) and n ∈ N, we 1 1 1 have T (yαn )(β, k, m) = km fβk (yαn ) = km fβk (Qα (yαn )) = km gβk Qβ Qα (yαn ) / 1 1 |fαk (yαn )| ≤ km yαn . (gβk ◦ Qβ + 1) = 0 for β = α. Also, |T (yαn )(α, k, m)| = km Thus T (yαn ) ∈ c0 (Γ). Since we have seen above that X = span{yαn ; α ∈ [ω, μ), n ∈ N}, it follows that T maps into c0 (Γ) and so X has a fundamental coordinate system.
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We proceed by deducing the result (v) from Corollary 9 and the result (ii) from Theorem 10. We start with an easy observation. Fact 7. Let X be a normed linear space and {(xγ ; fγ )}γ∈Γ ⊂ X × X ∗ . Then x ∈ ∗ span{xγ ; fγ (x) = 0} for every x ∈ X if and only if f ∈ spanw {fγ ; f (xγ ) = 0} for every f ∈ X ∗ . Proof. ⇒ Assume that it is not true for some f ∈ X ∗ . Denote A = {γ ∈ Γ; f (xγ ) = 0}. By the separation theorem there is x ∈ X such that f (x) = 0 and fγ (x) = 0 for each γ ∈ A. It follows that x ∈ span{xγ ; fγ (x) = 0} ⊂ span{xγ ; γ ∈ Γ \ A} ⊂ {f }⊥ , a contradiction. ⇐ Assume that it is not true for some x ∈ X. Denote A = {γ ∈ Γ; fγ (x) = 0}. By the separation theorem there is f ∈ X ∗ such that f (x) = 0 and f (xγ ) = 0 for each ∗ ∗ γ ∈ A. It follows that f ∈ spanw {fγ ; f (xγ ) = 0} ⊂ spanw {fγ ; γ ∈ Γ \ A} ⊂ {x}⊥ , a contradiction. 2 The first part of the next theorem is probably folklore among experts. We include the proof for the convenience of the reader. Theorem 8. Let X be a WCG Banach space and let K ⊂ X be a weakly compact convex symmetric set that generates the space X. Then X has a strong Markushevich basis {(xγ ; fγ )}γ∈Γ ⊂ K × X ∗ . Such a basis has the following properties: T (f ) = (f (xγ ))γ∈Γ ∗ is a bounded linear operator from X ∗ to c0 (Γ) and f ∈ spanw {fγ ; f (xγ ) = 0} for each f ∈ X ∗. Proof. We prove the first part by transfinite induction on dens X. Suppose first that X is separable. Let {zn }n∈N ⊂ K be a dense set in K and {hn }n∈N a norming set in X ∗ . Note that span{zn } = X. By [5, Theorem 4.59] there is a Markushevich basis {(yn ; gn )}n∈N of X such that span{yn } = span{zn } and span{gn } = span{hn }. In particular, this basis is norming. Hence by [9, Theorem 1.42] there is a strong Markushevich basis {(xn ; fn )}n∈N of X such that {xn } ⊂ span{yn } = span{zn }. Since span{zn } ⊂ n∈N nK, by scaling we may assume that {xn } ⊂ K. Now assume that dens X > ω and the statement is true for all WCG spaces of density less than dens X. By [5, Theorem 13.6] there is a PRI {Pα }α∈[ω,μ] on X such that Pα (K) ⊂ K for each α ∈ [ω, μ]. Denote Qα = Pα+1 − Pα . For each α ∈ [ω, μ) the space Qα (X) is of density at most card α < dens X and is generated by the weakly compact convex symmetric set 12 Qα (K). Thus by the inductive hypothesis Qα (X) has 1 α α a strong Markushevich basis {(xα γ ; gγ )}γ∈Γα such that {xγ }γ∈Γα ⊂ 2 Qα (K) ⊂ K. Put α α α α fγ = gγ ◦ Qα . We claim that {(xγ ; fγ )}α∈[ω,μ),γ∈Γα is a strong Markushevich basis of X. Indeed, Qα (xβη ) = Qα (Qβ (xβη )) = 0 and hence fγα (xβη ) = gγα (Qα (xβη )) = 0 for α = β. α α α α Further, fγα (xα η ) = gγ (Qα (xη )) = gγ (xη ) = δγ,η (the Kronecker delta). Fix any x ∈ X. α α Then Qα (x) ∈ span{xγ ; γ ∈ Γα : gγα (Qα (x)) = 0} = span{xα γ ; γ ∈ Γα : fγ (x) = 0}.
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α Hence x ∈ span{Qα (x); α ∈ [ω, μ)} ⊂ span α∈[ω,μ) span{xα γ ; γ ∈ Γα : fγ (x) = 0} ⊂ α span{xα γ ; α ∈ [ω, μ), γ ∈ Γα : fγ (x) = 0}. Finally, note that this strongness property implies that the biorthogonal system is total. To prove the second part of the theorem, denote by τ the topology on X ∗ given by the uniform convergence on K. Put T (f ) = (f (xγ ))γ∈Γ for f ∈ X ∗ . Then T is clearly a bounded linear operator from X ∗ to ∞ (Γ). Since T (f ) = supγ∈Γ |f (xγ )| ≤ supx∈K |f (x)|, the operator T is moreover τ - · continuous. Further, T (fα ) ∈ c00 (Γ) ∗ for every α ∈ Γ. By the Mackey–Arens theorem, spanτ {fα } = spanw {fα } = X ∗ . Consequently, T (X ∗ ) = T (spanτ {fα }) ⊂ span{T (fα )} ⊂ c0 (Γ). The rest follows from Fact 7. 2 We remark that the heart of the construction of a strong Markushevich basis lies in the separable case and is seriously difficult. The strongness of the PRI then arranges the rest. However, for our purpose (the second part of the previous theorem), the full strongness (and even the biorthogonality) of the Markushevich basis is not necessary. It would be sufficient to carry the required properties through the transfinite induction and use just the strongness provided by the PRI. The weak compactness is indispensable though. Corollary 9. Let X be a normed linear space such that X ∗ is WCG. Then X has a fundamental coordinate system. Proof. Let {(fγ ; Fγ )}γ∈Γ ⊂ X ∗ × X ∗∗ be a Markushevich basis from Theorem 8. Note w∗
that {fγ }γ∈Γ is bounded. Fix γ ∈ Γ. Then by the Goldstine theorem Fγ ∈ B for ∗ some ball B ⊂ X. Since X has the property C ([5, Definition 14.32, Theorem 14.31]), by [5, Theorem 14.37] there is a countable set of vectors {xγn }n∈N ⊂ B such that ∗ Fγ ∈ convw {xγn }n∈N . We claim that {(xγn ; n1 fγ )}γ∈Γ,n∈N is a fundamental coordinate system. Indeed, T (x) = n1 fγ (x) γ∈Γ,n∈N is a bounded linear operator from X to c0 (Γ × N), as (fγ (x))γ∈Γ ∈ c0 (Γ) by Theorem 8. Fix x ∈ X and denote A = {xγn ; fγ (x) = 0, n ∈ N}. ∗ Theorem 8 implies that F ∈ spanw {Fγ ; F (fγ ) = 0} for any F ∈ X ∗∗ and so ∗
x ∈ spanw {Fγ ; fγ (x) = 0} ⊂ spanw
∗
∗
∗
convw {xγn }n∈N = spanw A.
γ∈Γ : fγ (x) =0
But since x ∈ X and span A ⊂ X, this means that x ∈ spanw A = spanA. 2 We note that there is a Banach space X such that it is a second dual space, it has an equivalent C 1 -smooth norm, X ∗ is a subspace of a Hilbert-generated space (in particular a subspace of a WCG space), and there is no bounded linear one-to-one operator from X to c0 (Γ), [1]. Therefore there is no hope for generalising the result (v) beyond the dual being WCG using the approach above (or the original proof as well – both result in a linear injection into c0 (Γ)).
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Theorem 10. Every Banach space that belongs to a P-class has a fundamental coordinate system. Proof. Let X be a P-class and X ∈ X . We use transfinite induction on dens X. If X is separable, then we can use the existence of a strong Markushevich basis. However, this difficult result is not necessary. A direct construction is as follows: Let {yn }n∈N ⊂ X be dense in X and let {gn }n∈N ⊂ X ∗ be such that it separates the points of X and gn ≤ n1 . For k, n ∈ N put xkn = yn and fkn = n1 gk . Then {(xkn , fkn )}k,n∈N is a fundamental 1 coordinate system: Fix x ∈ X. Then |fkn (x)| ≤ nk x. Also, there is m ∈ N such that gm (x) = 0 and so x ∈ span{yn ; n ∈ N} = span{xmn ; n ∈ N} ⊂ span{xkn ; fkn (x) = 0}. Now assume that dens X > ω and every space in X of density less than dens X has a fundamental coordinate system. Let {Pα }α∈[ω,μ] be a PRI on X such that Pα (X) ∈ X for α ∈ [ω, μ). Put Qα = Pα+1 − Pα . By the inductive hypothesis, for each α ∈ [ω, μ) there α is a fundamental coordinate system {(xα γ ; gγ )}γ∈Γα on Pα (X) and there is Kα > 0 such 1 α that {gγ }γ∈Γα ⊂ B(0, Kα ). Since Qα (X) ⊂ Pα+1 (X), we may set fγα+1 = Kα+1 gγα+1 ◦Qα and note that fγα+1 ≤ 2. We claim that {(xα+1 ; fγα+1 )}α∈[ω,μ),γ∈Γα+1 is a fundamental γ coordinate system on X. Indeed, the formula T (x) = (fγα+1 (x))α∈[ω,μ),γ∈Γα+1 clearly defines a bounded lin ear operator from X to ∞ (Γ), where Γ = α∈[ω,μ) {α} × Γα+1 . Now fix x ∈ X and ε > 0. Then the set A = {α ∈ [ω, μ); Qα (x) > ε} is finite. So, |fγα+1 (x)| ≤ 1 α+1 Qα (x) ≤ ε whenever α ∈ [ω, μ) \ A and γ ∈ Γα+1 . On the other hand, if Kα+1 gγ
α ∈ A, then the set {γ ∈ Γα+1 ; |fγα+1 (x)| > ε} = γ ∈ Γα+1 ; |gγα+1 (Qα (x))| > Kα+1 ε is finite by the definition of a fundamental coordinate system. Finally, as Qα (x) ∈ Pα+1 (X), the assumption gives Qα (x) ∈ span{xα+1 ; gγα+1 (Qα (x)) = 0} = span{xα+1 ; fγα+1 (x) = 0}. γ γ α+1 α+1 Therefore x ∈ span{Qα (x); α ∈ [ω, μ)} ⊂ span α∈[ω,μ) span{xγ ; fγ (x) = 0} ⊂ span{xα+1 ; fγα+1 (x) = 0}. 2 γ References [1] Spiros A. Argyros, Sophocles Mercourakis, A note on the structure of WUR Banach spaces, Comment. Math. Univ. Carolin. 46 (3) (2005) 399–408. [2] Robert Deville, Gilles Godefroy, Václav Zizler, The three space problem for smooth partitions of unity and C(K) spaces, Math. Ann. 288 (1) (1990) 613–625. [3] Robert Deville, Gilles Godefroy, Václav Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monogr. Surv. Pure Appl. Math., vol. 64, Longman Scientific & Technical, Harlow, 1993. [4] Marián Fabian, Gâteaux Differentiability of Convex Functions and Topology – Weak Asplund Spaces, CMS Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1997. [5] Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, Václav Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books Math., Springer, New York, 2011. [6] Gilles Godefroy, Stanimir L. Troyanski, John H.M. Whitfield, Václav Zizler, Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (3) (1983) 344–352. [7] Petr Hájek, Richard Haydon, Smooth norms and approximation in Banach spaces of the type C(K), Q. J. Math. 58 (2) (2007) 221–228. [8] Petr Hájek, Michal Johanis, Smooth Analysis in Banach Spaces, De Gruyter Ser. Nonlinear Anal. Appl., vol. 19, Walter de Gruyter, Berlin, 2014. [9] Petr Hájek, Vicente Montesinos, Jon Vanderwerff, Václav Zizler, Biorthogonal Systems in Banach Spaces, CMS Books Math., vol. 26, Springer, New York, 2008.
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[10] Richard Haydon, Smooth functions and partitions of unity on certain Banach spaces, Q. J. Math. 47 (4) (1996) 455–468. [11] Jerzy Kąkol, Wiesław Kubiś, Manuel López-Pellicer, Descriptive Topology in Selected Topics of Functional Analysis, Dev. Math., vol. 24, Springer, New York, 2011. [12] David P. McLaughlin, Smooth partitions of unity in preduals of WCG spaces, Math. Z. 211 (1992) 189–194.