Anti M-Weierstrass function sequences

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is an accepted manuscript of a paper published in: Journal of Mathematical Analysis and Applications (2020): 1 November DOI: 10.1016/j.jmaa.2020.124261 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This is an Accepted Manuscript of an article published by Elsevier in [JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS] on [1 November 2020], available at: https://doi.org/10.1016/j.jmaa.2020.124261].” ” ANTI M-WEIERSTRASS FUNCTION SEQUENCES MAR´ IA DEL CARMEN CALDER´ ON-MORENO, PABLO JOS´ E GERLACH-MENA, AND JOS´ E ANTONIO PRADO-BASSAS Abstract. Large algebraic structures are found inside the space of sequences of continuous functions on a compact interval having the property that, the series defined by each sequence converges absolutely and uniformly on the interval but the series of the upper bounds diverges. So showing that there exist many examples satisfying the conclusion but not the hypothesis of the Weierstrass M-test. 1. Introduction In any field of Mathematics, whenever the uniform convergence of a series of functions P∞ n=1 fn(x) must be studied (where fn:X→K,X6=∅and K= the real line Ror the complex plane C), the tool we first think about is the well-known Weierstrass M-test. If there exists a sequence (Mn)nof positive numbers such that its series is convergent and |fn(x)| ≤ Mnfor every x∈Xand every n∈N(:=the set of all positive integers), then the series P∞ n=1 fn(x) is absolutely and uniformly convergent on X(see, for instance, [2, § 9.6]). The converse is false in general. Concretely, the hypothesis about the majorant sequence can be dropped without losing the uniform convergence of the series. Indeed, for every n∈Nand every x∈[0,1], define (1) fn(x) = (1 nsin2(2n+1πx) if x∈2−(n+1),2−n 0 elsewhere. The sequence fn(x) cannot be majorated by any summable sequence of positive numbers, since supx∈[0,1] |fn(x)|=1 n; however, the series P∞ n=1 fn(x) is absolutely and uniformly convergent (see [16, Chapter 1, Example 10] for more details). Such a sequence of functions will be called Anti M-Weierstrass (see Definition 1.1 below). We denote by C([a, b])Nthe set of all sequences of continuous functions on a nondegenerate compact interval [a, b]⊂R. Recall that if we endow C([a, b])Nwith the Key words and phrases. series of functions, uniform convergence, divergence, spaceability, algebrability. 2020 Mathematics Subject Classification. 46B87, 28A20, 40A30, 46A45. The authors have been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 and by MICINN Grant PGC2018-098474-B-C21. 1 2 CALDER ´ ON, GERLACH, AND PRADO product topology inherited from (C([a, b]),k·k∞), where kfk∞= supx∈[a,b]|f(x)| is the classical supremum norm, we get a complete metrizable and separable topological vector space (see [18, Chapter 7]). Definition 1.1. Let (fn)n∈C([a, b])Nbe a sequence of continuous functions on [a, b]. We say that (fn)nis an Anti M-Weierstrass sequence if P∞ n=1 fn(x) is absolutely and uniformly convergent on [a, b], but P∞ n=1 kfnk∞diverges. We will denote by AMW([a, b]) (or just AMW, whenever there is no possibility of confusion) the family of Anti M-Weierstrass sequences on [a, b]. The aim of this paper is constructing large linear structures of Anti M-Weiers- trass sequences, and this objective lies in the branch of Lineability, a trend of research that has attracted the attention of many mathematicians during the past few decades. We introduce in the next paragraph the main notions about Lineability we are going to deal with. We refer the interested reader to [4, 15] for a general background. Given a vector space Xand a (finite or infinite) cardinal number κ, we say that a subset Aof Xis κ-lineable whenever there is a vector space Mof dimension κ such that M\ {0} ⊂ A; when κ= dim(X), we speak about maximal-lineability. Moreover, if Xis a topological vector space, then Ais said to be spaceable (denselineable, κ-dense-lineable, maximal-dense-lineable, resp.) in X, if there is a closed infinite dimensional (a dense, a dense κ-dimensional, a dense dim(X)-dimensional, resp.) vector space Mwith M\ {0} ⊂ A. When, in addition, Xis contained in some (linear) algebra, then Ais called κ-algebrable if there is an algebra Mso that M\ {0} ⊂ Aand the cardinality of any minimal system of generators of Mis κ; if the algebra can be taken to be free, we say that Ais strongly κ-algebrable; and if the (free) algebra is dense in X, we speak about (strongly) κ-dense-algebrability. If the algebraic structure of Xis commutative, the strong algebrability is equivalent to the existence of a generating system Bwith card(B) = κof the algebra Msuch that for any N∈N, any nonzero polynomial Pin Nvariables without constant term and any distinct b1, . . . , bN∈B, we have that P(b1, . . . , bN)∈M\ {0}. Concerning lineability of series, in 2005 Bayart [10] showed that the set of continuous functions on the unit circle Twhose Fourier series diverges on a set E⊂T of measure zero is dense-lineable. A year later, Aron, P´erez-Garc´ıa and Seoane- Sep´ulveda [6] established the dense-algebrability. When Eis countable, results on lineability of divergent Fourier series with additional properties are obtained in [13] or [17]. Again Bayart [9, 10] showed that the set of Dirichlet series f(s) = P∞ n=1 ann−s that are bounded in the right half-plane and diverge everywhere in the imaginary axis is lineable and spaceable. Together with Quarta [11], they were able to establish also the algebrability. In the setting of Banach spaces, Aizpuru, P´erez-Eslava and Seoane-Sep´ulveda in 2006 [1], asserted that the set of unconditionally convergent but not absolutely ANTI M-WEIERSTRASS FUNCTION SEQUENCES 3 convergent series of any infinite dimensional Banach space is c-lineable (where c denotes the cardinality of continuum). Moreover, the set of sequences such that its partial sums are bounded but the series diverges is also c-lineable. If we focus on the complex plane, Bartoszewicz, G lab and Poreda proved in [8] that the set of nonabsolutely convergent complex series and the set of divergent complex series with bounded partial sums are c-algebrable. Following with sequences of real numbers, in 2013 Bartoszewicz and G lab [7] showed that c0\Sp≥1`pis densely strongly c-algebrable, where c0denotes the set of all real sequences converging to zero, and `pis the set of all p-summable real sequences. Later, in 2017, Ara´ujo et al. [3] studied the linear structure of the family of sequences of real numbers whose corresponding series fail both the root and ratio test. Specifically, they show that the set of sequences in `1generating series for which the ratio or the root tests fail and the set of sequences in ω(the vector space of all real sequences endowed with the product topology) generating divergent series for which the ratio and the root tests fail are c-dense-lineable in `1or ω. In this paper we are concerned with lineability –in its several degrees– of the family AMW described in Definition 1.1. In Section 2 we shall define a special family of sequences of functions that will be crucial in the forthcoming constructions and provide several auxiliar results about it. Section 3 is devoted to a construction of several vector spaces inside AMW and to get its maximal dense lineability and spaceability. Finally, in Section 4, we conclude the paper by establishing, as a consequence of a more general result, the strong c-algebrability of our family AMW. 2. Concepts and Examples Inspired by the Example (1) of Anti M-Weierstrass sequence provided in the introduction, we can establish a very useful auxiliary result. By supp(f) we denote the support of a function f: [a, b]→K, that is, the set of points x∈[a, b] where fdoes not vanish. Let Fbe the family of all sequences of functions (un)n∈C([a, b])Nsuch that: •The supports are pairwise disjoint, that is, (2) supp(un)∩supp(um) = ∅, n 6=m. •The sequence (un)nis both uniformly bounded and uniformly far from zero, that is, (3) 0 <inf n||un||∞≤sup n ||un||∞<+∞. Lemma 2.1. Let (un)n∈ F and let (an)n⊂R. Then we have: (a) The series ∞ X n=1 anun(x)converges absolutely on [a, b]. 4 CALDER ´ ON, GERLACH, AND PRADO (b) The series ∞ X n=1 anun(x)converges uniformly on [a, b]if and only if (an)n∈ c0. (c) The series ∞ X n=1 kanunk∞<+∞if and only if (an)n∈`1. Proof. (a) The absolute convergence of the series is immediate, since the disjointness of the supports of the un’s implies that, for a fixed x0∈[a, b], either un(x0)=0 for all n∈N, or there exists only one n0∈Nsuch that x0∈supp(un0), and ∞ X n=1 |anun(x0)|=|an0un0(x0)|(<+∞). (b) Firstly, assume that (an)n∈c0. Because (un)n∈ F, then M:= supn||un||∞∈ (0,+∞). Given any ε > 0, there exists N∈Nsuch that |an|<ε Mfor any n≥N. Thus, as for each x∈[a, b] there is at most one n0≥Nsuch that x∈supp(un0),  ∞ X n=N anun(x) =|an0un0(x)| ≤ M·ε M=ε. Hence, the uniformly convergence of the series on [a, b] is obtained. Conversely, as the series is uniformly convergent, we have anun(x)→0 (n→ ∞) uniformly on [a, b]. Because (un)n∈ F, then L:= infn||un||∞>0. Given any ε > 0, there is n0∈Nsuch that |anun(x)|< ε ·Lfor any x∈[a, b] and any n≥n0. But for any n∈N, there exists, by continuity, a point xn∈[a, b] such that |un(xn)|=||un||∞≥L, hence for n≥n0we get |an|=|anun(xn)| |un(xn)|≤ε·L L=ε, and an→0 (n→ ∞). (c) As (un)n∈ F, we have (4) 0 < L := inf n||un||∞≤M:= sup n ||un||∞<+∞. Thus, by the comparison test (an)n∈`1if and only if (||anun||∞)n∈`1, and we have (c).  As a consequence of the above lemma we obtain the next fact. Corollary 2.2. For any sequence of functions (un)n∈ F and any sequence of scalars (an)n∈c0\`1, the sequence of continuous functions (anun)nis in AMW. In particular, this corollary will allow us to provide a wide plethora of sequences exhibiting this strange behaviour. In the following example we show that from any non null function f∈C([a, b]), many sequences in Fcan be constructed. ANTI M-WEIERSTRASS FUNCTION SEQUENCES 5 Example 2.3. Let f∈C([a, b]) \ {0}. Let Λ := (αn)nbe any sequence of scalars such that a=α1< α2<· · · < αn−1< αn<· · · −→ b(n→ ∞). For each n∈Nwe define the function uΛ,f n: [a, b]→Rby uΛ,f n(x) :=                      f(a)·x−α3n−2 α3n−1−α3n−2 if x∈[α3n−2, α3n−1] fa+b−a α3n−α3n−1 (x−α3n−1)if x∈[α3n−1, α3n] f(b)·α3n+1 −x α3n+1 −α3n if x∈[α3n, α3n+1] 0 otherwise. It is clear that uΛ,f n∈C([a, b]) and supp(uΛ,f n)⊂(α3n−2, α3n+1) for each n∈N, so the supports of the uΛ,f n’s are pairwise disjoint. In addition, ||uΛ,f n||∞=||f||∞∈ (0,+∞) for any n∈N. So, trivially, (uΛ,f n)n∈ F. Therefore, for any prefixed increasing sequence of scalars Λ as above, we have defined an injective mapping JΛ:C([a, b]) \ {0} −→ F f7−→ JΛ(f) := (uΛ,f n)n, satisfying the following properties: (1) uΛ,f n(α3n−1) = f(a) and uΛ,f n(α3n) = f(b) for any n∈Nand any f∈ C([a, b]) \ {0}. (2) supp(uΛ,f n)⊂(α3n−2, α3n+1) for any f∈C([a, b]) \ {0}and any n∈N. (3) For any n∈Nthere exists a linear affine transformation τnsuch that τn([α3n−1, α3n]) = [a, b] and uΛ,f n=f◦τnon [α3n−1, α3n] for each f∈ C([a, b]) \ {0}. (4) ||uΛ,f n||∞=||f||∞for any n∈Nand any f∈C([a, b]) \ {0}. Observe that if we define JΛ(0) as the null sequence, we have that JΛis a linear and injective mapping from C([a, b]) to F ∪{0}. In particular if Cis a linear vector subspace in C([a, b]) with dimension κ, then JΛ(C) is a linear vector subspace in F ∪ {0}with dimension κ. Furthermore, let `∞(C([a, b])) be the vector space of all uniformly bounded sequences of continuous functions in [a, b], which is a Banach space when endowed with the uniform supremum norm ||(fn)n||`∞:= sup n ||fn||∞. Observe that F ⊂ `∞(C([a, b])), and for any sequence Λ = (αn)nas above and for each function f∈C([a, b]) we have JΛ(f)∈`∞(C([a, b])) and ||JΛ(f)||`∞= sup n ||uΛ,f n||∞=||f||∞. 6 CALDER ´ ON, GERLACH, AND PRADO Thus, we have obtained the following useful statement. Proposition 2.4. Let Λ = (αn)n⊂Rwith a=α1< α2<· · · < αn−1< αn< · · · −→ b(n→ ∞).Then the mapping JΛ:C([a, b]) −→ `∞(C([a, b])) f7−→ JΛ(f) := (uΛ,f n)n, is a linear isometry whose range is contained in F ∪{0}. In particular, the normed spaces (C([a, b]),|| · ||∞)and (JΛ(C([a, b])),|| · ||`∞)are isometrically isomorphic. 3. Lineability and spaceability in AMW In this section we show that AMW is maximal dense lineable but at the same time we provide many examples of concrete linear structures in AMW. Theorem 3.1. Let M be a linear subspace such that M\{0} ⊂ c0\`1and dim(M) = κ. Let Ibe a set with card(I) = κand {(ai n)n}i∈Ibe an algebraic basis of M. Then for any sequence of functions (un)n∈ F, the set M:= {(anun)n: (an)n∈M} is a linear subspace of dimension κ, the family {(ai nun)n}i∈Iis an algebraic basis of Mand M\{0} ⊂ AMW. Proof. It is clear that the mapping Φ : (an)n∈c07−→ (anun)n∈C([a, b])Nis linear. Moreover, the fact that un6= 0 for all n∈Nimplies that Φ is one-to-one. Then all conclusions follow from the equality M= Φ(M) and from Corollary 2.2.  By following a dual way (that is, fixing this time a sequence of c0\`1), we are able to obtain subspaces of AMW. Theorem 3.2. Let (an)n∈c0\`1with an6= 0 for all n∈N. Let Ube a linear subspace of dimension κ, having an algebraic basis {(ui n)n}i∈Iwith card(I) = κ, such that U\ {0}⊂F. Then U:= {(anun)n: (un)n∈U} is a linear subspace of dimension κ, the family {(anui n)n}i∈Iis an algebraic basis of U, and U \ {0} ⊂ AMW. Proof. It is similar to the proof of Theorem 3.1, except that this time one considers the mapping Ψ : (un)n∈U7−→ (anun)n∈C([a, b])N. Then Ψ is linear and injective (because an6= 0 for each n∈N), U= Ψ(U), and Corollary 2.2 applies again.  Recall that in any separable, metrizable, complete topological vector space (as it is the case of c0,`p,C([a, b]) or C([a, b])N) the maximal dimension of any linear vector subspace is the dimension of the continuum c. In particular if we consider ANTI M-WEIERSTRASS FUNCTION SEQUENCES 7 in Theorem 3.1 any linear vector subspace Min c0\`1of dimension c, for instance M= span 1 ncn:c∈(0,1)), we obtain: Corollary 3.3. The set AMW is maximal lineable in C([a, b])N. We can obtain the same from Theorem 3.2 just considering the linear subspace in F ∪ {0}given by U= span{JΛ(xc) : c∈(0,+∞)}, where Λ is any prefixed sequence of scalars as in Example 2.3. We denote by c0(C([a, b])) the topological vector space of all sequences (fn)n∈ C([a, b])Nsuch that ||fn||∞→0 (n→ ∞). It is known that c0(C([a, b])) endowed with the topology inherited by ||(fn)n||`∞:= supnkfnk∞is a separable Banach space (indeed, for the separability just take the set of sequences (gn)nsuch that gn6= 0 for finitely many terms which belong to a dense and numerable set of C([a, b]) -for example, polynomials with rational coefficients-). It is clear that AMW ⊂ c0(C([a, b])), so if we take into account the topological structure of c0(C([a, b])), we can focus our attention on density properties enjoyed by AMW. A first positive result in this direction is given by the application of the following lemma due to Bernal [12] (see also [4, Chapter 4], [5] and [14]). Lemma 3.4. Let Xbe a separable metrizable topological vector space, A⊂X maximal lineable and B⊂Xdense lineable in Xwith A∩B=∅. If A+B⊂A then Ais maximal dense lineable in X. Theorem 3.5. The family AMW of Anti-M Weierstrass sequences is maximal dense lineable in c0(C([a, b])). Proof. From Corollary 3.3, the family AMW is maximal lineable. Now, the family c00(C([a, b])) := {(fn)n∈C([a, b])N: there is Nsuch that fn= 0 for any n≥N} is a dense linear subspace of c0(C([a, b])) and so, trivially, it is a dense lineable subset of c0(C([a, b])). For a fixed (fn)n∈ AMW, each sequence (gn)n∈c00(C([a, b])) only modifies, under addition, a finite number of components of (fn)n. So (fn+ gn)n∈ AMW. In other words, c00(C([a, b])) + AMW ⊂ AMW. Now, an application of Lemma 3.4, with X:= c0(C([a, b])), A:= AMW and B:= c00(C([a, b])), gives us the maximal dense-lineability of AMW in c0(C([a, b])).  Finally, we are also able to get the spaceability of the anti-M Weierstrass family of sequences. For this, the linear isometric property of the mapping JΛand the (trivial) spaceability of C([a, b]) (see [4], [15]) play an important role. Theorem 3.6. The family AMW of anti-M Weierstrass sequences is spaceable in c0(C([a, b])). In fact, for any Λ=(αn)nwith a=α1< α2<· · · < αn<· · · → b(n→ ∞), any sequence (an)n∈c0\`1and any infinite dimensional closed vector subspace Cof C([a, b]), the set M:= {(anuΛ,f n)n:f∈ C} 8 CALDER ´ ON, GERLACH, AND PRADO is an infinite dimensional closed vector subspace of (c0(C([a, b])),||·||`∞)such that M\{0} ⊂ AMW. Proof. Let Λ, (an)nand Cas above. It is clear that Mis a linear space. From Proposition 2.4, JΛis a linear isometry, so JΛ(C) is an infinite dimensional closed vector space of `∞(C([a, b])) such that JΛ(C)\ {0} ⊂ F. But (an)n∈c0, so there exists L > 0 such that |an| ≤ Lfor all n∈N, and for any f∈ C we have ||(anuΛ,f n)n||`∞= sup n∈N ||anuΛ,f n||∞≤L· ||(uΛ,f n)n||`∞, and ||anuΛ,f n||∞=|an| · ||f||∞→0 (n→ ∞). Therefore Mis a closed vector subspace of (`∞(C([a, b])),||·||`∞) that is contained in c0(C([a, b])). But c0(C([a, b])) is closed in `∞(C([a, b])), hence Mis a closed vector subspace of (c0(C([a, b])),|| · ||`∞). Of course, Mis infinite dimensional because Cis, there are infinitelly many an6= 0 (recall (an)n/∈`1) and JΛis a linear isometry such that uΛ,f n= 0 for some n∈Nif and only if f= 0 on [a, b]. Finally, as (an)n∈c0\`1and JΛ(C)\ {0}⊂F, we get from Corollary 2.2, M\{0} ⊂ AMW. Remark 3.7. Observe that, for any sequence Λ as above and any (an)n∈c0\`1 we have that for L:= maxn∈N|an| ∈ (0,+∞), ||λ1(anuΛ,f1 n)n+· · · +λs(anuΛ,fs n)n||`∞=||(anuΛ,(λ1f1+···+λsfs) n)n||`∞ =L· ||λ1f1+· · · +λsfs||∞, where λ1, . . . , λs∈Rand f1, . . . , fs∈C([a, b]). In particular, we can drop the restriction an6= 0 in Theorem 3.2 and we can state the algebraic basis of the subspaces in AMW. Specifically: (1) If {fi}i∈I⊂C([a, b]) are linearly independent, then the family M={(anuΛ,fi n)n: i∈I}is linearly independent and span(M)\ {0} ⊂ AMW; (2) If span{fi:i∈I}is a closed vector subspace of C([a, b]) with dimension κ, then the closed vector subspace M:= span{(anuΛ,fi n)n:i∈I}of c0(C([a, b])) has the same dimension κ, and M\{0} ⊂ AMW. 4. Algebrability of AMW In this section we provide several ways to generate free algebras in AMW. But before this we state two technical lemmas. Lemma 4.1. Let Ube a free algebra in C([a, b]), generated by U:= {ui}i∈I. Then, for any family P={pi}i∈Iof polynomials of degree exactly 1, the set UP:= {pi◦ui}i∈Iis a generator system of a free algebra in C([a, b]).