Algebraic structure of continuous, unbounded and integrable functions

Algebraic structure of continuous, unbounded and integrable functions M.C. Calder´on-Moreno1, P.J. Gerlach-Mena1, J.A. Prado-Bassas1,∗ Dpto. An´alisis Matem´atico, Fac. Matem´aticas, Universidad de Sevilla. Avda. Reina Mercedes s/n, 41080 Sevilla, Spain. Abstract In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in [0,+∞) and of the family of sequences of these functions converging to zero uniformly on compacta and in L1-norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero. Keywords: Continuous unbounded functions, integrable functions, lineability, algebrability. 2010 MSC: 15A03, 26A15, 46E30 1. Introduction Contrary to what happens with a series of real numbers, there are even integrable functions on [0,+∞) that do not converge pointwise to zero as x→+∞. In fact, it is easy to construct unbounded, continuous and integrable functions on [0,+∞) (see Example 2.1). In this paper we will analyze the existence of large algebraic structures of sets of such functions and of sequences of these functions converging to zero, in a manner to be specified later. ∗Corresponding author Email addresses: [email protected] (M.C. Calder´on-Moreno), [email protected] (P.J. Gerlach-Mena), [email protected] (J.A. Prado-Bassas) 1The authors have been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127, Grant P08-FQM-03543 and by MINECO Grant MTM2015- 65242-C2-1-P. Preprint submitted to Journal of Mathematical Analysis and ApplicationsSeptember 28, 2018 We will first establish some notation. As is standard, we will denote by Nand Rthe set of positive integers and the real line, respectively. For any interval I⊂[0,+∞), by C(I) we will denote the vector space of all continuous functions on I; in the special case I= [0,+∞), the space C([0,+∞)) will be endowed with the topology of the uniform convergence on compacta, under which C([0,+∞)) becomes a complete metrizable topological vector space or F-space. Also, if I⊂[0,+∞) is any interval, L1(I) will stand for the space of all Lebesgue integrable functions on I, endowed with its usual topology of the L1-norm kfkI=RI|f(x)|dx. The general aim of this paper is the search for linear structures within a nonlinear setting, a trending branch that has attracted the attention of many mathematicians within the last decade. The concept of lineability was coined by the Russian mathematician V.I. Gurariy [17]. He later partnered with R.M. Aron, J.B. Seoane-Sep´ulveda et al. [5], the concept of algebrability (see [6]), which relates to the algebraic size of a set. We refer to the survey [12] and the book [4] for a wide background on this topic. Specifically, we introduce the next definitions that will be used later. Given a vector space Xand a subset A⊂X, we say that Ais: •lineable whenever there is an infinite dimensional vector space Msuch that M\ {0} ⊂ A; •α-lineable if, in addition, dim(M) = α; •maximal lineable if Ais dim(X)-lineable. Moreover, if Xis a topological vector space, then Ais said to be: •dense-lineable if there is a dense vector space Mwith M\ {0} ⊂ A; •maximal dense-lineable if we also have that dim(M) = dim(X). When Xis a topological vector space contained in some (linear) algebra, then Ais called: •algebrable if there is an algebra Mso that M\ {0} ⊂ Aand Mis infinitely generated, that is, the cardinality of any system of generators of Mis infinite; •strongly algebrable if the algebra Mabove can be taken free. 2 If the algebraic structure of Xis commutative, the strong algebrability is equivalent to the existence of a generating set Bof the algebra Msuch that for any s∈N, any nonconstant polynomial Pin svariables and any distinct f1, . . . , fs∈B, we have P(f1, . . . , fs)∈M\ {0}. Focusing our attention on lineability of continuous functions, Gurariy [17] stated the lineability of the set of functions that are continuous on [0,1] but nowhere differentiable. Later in 2013 (see [18]), Jim´enez-Rodr´ıguez, Mu˜noz- Fern´andez and Seoane-Sep´ulveda gave the first constructive proof of the clineability of the so called Weierstrass’ Monsters, while Bayart and Quarta [7] proved the dense algebrability (that is, the existence of a dense infinitely generated algebra) of the more restrictive family of continuous and nowhere H¨older functions on [0,1]. In the same vein, the first and third author, jointly with Bernal [9], proved the maximal lineability and strong algebrability of the set of continuous functions ϕ: [0,1] →R2whose image has nonempty interior and a zero-(Lebesgue) measure boundary as well as the maximal dense-lineability of the family of continuous functions from [0,1] to R2whose images have positive Lebesgue measure. As a related result, in 2014 Albuquerque [1] states the maximal lineability of the set of continuous surjections from Rto R2. Extensions of these results can be found in [2, 11]. But a large linear structure cannot always be obtained for continuous functions with additional properties. In 2004, Gurariy and Quarta showed that the set of continuous functions on [0,1] which admit one and only one absolute maximum is 1-lineable but not 2-lineable. In addition, they also proved that the set of continuous functions on the whole real line satisfying the mentioned property is 2-lineable, while the corresponding subset of continuous functions vanishing at infinity is not 3-lineable. Turning now to the framework of integrable functions, Mu˜noz, Palmberg, Puglisi and Seoane [20] proved the c-lineability of Lp[0,1]\Lq[0,1] for 1 ≤p < qand of Lp(I)\Lq(I) for p > q ≥1 and any unbounded interval I⊂R. In [5, 8, 10, 11, 13, 14, 19] some generalizations and extensions of these results are proved. Concerning the relationship between lineability and integrability, Garc´ıa, Mart´ın and Seoane showed in 2009 [16] that, given an arbitrary unbounded interval I⊂R, the set of all almost everywhere continuous bounded functions on Iwhich are also not Riemann integrable, contains an infinitely generated closed subalgebra. Moreover, they also proved the existence of an infinite dimensional closed vector space consisting (except for zero) of con- 3 tinuous and bounded functions on an unbounded interval Iwhich are not Riemann integrable; in particular, this set is c-lineable. In addition, they also state the lineability of the set of Riemann integrable functions (on an unbounded interval) which are not Lebesgue integrable and the c-lineability (in fact, the existence of a closed infinite dimensional subspace) of the set of Lebesgue integrable functions that are not Riemann integrable. In this direction, Bernal and Ord´o˜nez [11] show that the set of continuous and Riemann integrable functions on [0,+∞) which do not belong to Lp([0,+∞)) for any 0<p<+∞is maximal dense-lineable. On the other hand, Garc´ıa, Grecu, Maestre and Seoane [15] were able to show the existence of an infinite dimensional Banach space of bounded and Lebesgue integrable functions in [a, b] with antiderivatives at every point of the interval but, except for zero, are not Riemann integrable. Finally, in the setting of sequences of functions, there are, up to the authors knowledge, only few and relatively recent results. In fact, in 2014, Bernal and Ord´o˜nez proved in [11] that the family of sequences (fn)n⊂ RRNof continuous bounded and integrable functions such that kfnk∞ n→∞ −→ +∞, sup{kfnkL1:n∈N}<+∞but kfnkL16→ 0 (n→ ∞) is maximal lineable. Later, in 2017, Ara´ujo, Bernal, Mu˜noz, the third author and Seoane [3] were able to show the c-lineability of the family of sequences of Lebesgue measurable functions R→Rsuch that fnconverges pointwise to zero and fn(I) = Rfor any non-degenerate interval I⊂Rand any n∈N. Additionaly, the maximal dense-lineability (in the vector space of sequences of Lebesgue measurable functions [0,1] →R) of the family of sequences of Lebesgue measurable functions such that fn→0 in measure but not almost everywhere in [0,1] is proven. The aim of this paper is to contribute to the study of lineability and algebrability of continuous and integrable functions and of sequences of them. For that purpose, in Section 2 we focus on the linear and algebraic size of the set of continuous, unbounded and integrable functions on [0,+∞), paying special attention to the natural topology carried by the framework space. In Section 3, we turn to the problem of the algebraic genericity of the set of sequences of functions in the just mentioned class that tend to zero both in L1-norm and uniformly on compacta. We conclude the paper with a final section devoted to extend some results in the preceding sections by looking for a stronger way of convergence of sequences, the possible rate of growth of the functions, or the smoothness of them. 4 2. Unbounded, continuous and integrable functions As we said in the Introduction, in the context of continuous and integrable functions on [0,+∞), intuition leads us to believe that these functions should go somehow to zero when xtends to infinity. This is clearly true if fis decreasing or if it exists lim x→+∞f(x), but in general it is not difficult to provide examples of continuous and integrable functions on [0,+∞) that not only are not close to zero but also satisfy lim sup x→+∞ |f(x)|= +∞. The next example shows a classical undergraduate construction of an unbounded, continuous and integrable function on [0,+∞) that will be very useful going forward. Example 2.1. For any n∈N, we consider the triangular function Tn: [0,+∞)−→ Rgiven by: Tn(x) =      n(2n+1x+ (1 −n2n+1)) if x∈n−1 2n+1 , n, n(−2n+1x+ (1 + n2n+1)) if x∈n, n +1 2n+1 , 0 otherwise, and the function f: [0,+∞)−→ Rdefined by joining the previous triangles: f(x) = ∞ X n=1 Tn(x). That is, f“draws” all the triangles of height n, supported on intervals centered at nwith length 1 2n(n∈N). The function fis continuous by construction, and also integrable. Indeed, kfkL1=Z∞ 0 |f(x)|dx = ∞ X n=1 Z∞ 0 Tn(x)dx = ∞ X n=1 n 2n+1 <+∞. Furthermore, fis unbounded just by considering the sequence xn=n: f(xn) = f(n) = nn→∞ −→ +∞. Observe that with a similar construction we could get the value kfkL1∈ (0,+∞) arbitrarily small, just adjusting the size of the bases of the triangles. 5 From now on we denote by Athe family of unbounded continuous integrable functions in [0,+∞), that is, A=f∈ C([0,+∞)) ∩L1([0,+∞)) : lim sup x→+∞ |f(x)|= +∞. The natural question that now arises is how many functions are there in Aand what the meaning of “many” would be. Observe that Ais not a vector space, just take f(x) and e−x−f(x), where f∈ A. Observe also that the maximal dimension of a vector space M⊂ A ∪ {0}is c, where c=the dimension of continuum. In order to study the algebraic size of the set of these functions we need the next result that will allow us to get dense-lineability. Recall that given two sets A, B of a vector space, then Ais said to be stronger than Bif A+B⊂A(see [5, 11]). Lemma 2.2. Let Xbe a metrizable topological vector space, A⊂Xmaximal lineable and B⊂Xdense-lineable in Xwith A∩B=∅. If Ais stronger than Bthen Ais maximal dense-lineable. Theorem 2.3. The family Ais maximal lineable. Proof. We are looking for a vector subspace Mof dimension csuch that M⊂ A ∪ {0}. Consider the triangular function fgiven in the Example 2.1 and for any t∈0,1 8we define the functions: ft(x) := f(x−t). We have ft∈ A for each t∈0,1 8. Define Mas the set M:= span ft:t∈0,1 8. We claim that Mis our searched vector subspace. Let us assume that there are 0 < t1< t2< . . . < ts<1 8and scalars c1, c2, . . . , cs∈Rnot all simultaneously zero such that c1ft1(x) + c2ft2(x) + . . . +csfts(x) = 0 (x∈[0,+∞)). Without loss of generality we can assume c16= 0. But then, for any x0∈ 3 4+t1,3 4+t2, we get c1ft1(x0) + c2ft2(x0) + . . . +csfts(x0) = c1ft1(x0)6= 0. 6 So, the set {ft:t∈[0,1 8)}is linearly independent and dim(M) = c. It is obvious that every finite combination of elements of Mis continuous and integrable, hence it only remains to prove the unboundedness of each mentioned linear combination c1ft1+. . . +csfts. Note that we can assume cs6= 0. Since ts> tifor all i= 1,2, . . . , s −1, there is N∈Nsuch that the support of the triangles of ftsis disjoint with the support of the rest of triangles of fti. Indeed, there is N∈Nsuch that min{ts−ti:i= 1, . . . , s −1}>1 2N. So N+1 2N+1 +ti< N −1 2N+1 +ts, for any i= 1, . . . , s −1. Thus, by taking for any n>N the point xn:= n+ts, we have |c1ft1(xn) + c2ft2(xn) + . . . +csfts(xn)|=|cs|nn→∞ −→ +∞. Hence the family Ais maximal-lineable. Recall that X:= C([0,+∞)) ∩L1([0,+∞)) is a topological vector space when we endowed it with the natural translation-invariant distance given by dX(f, g) := kf−gkL1+ ∞ X n=1 1 2n·kf−gk∞,n 1 + kf−gk∞,n ,(1) where kfk∞,n = max{|f(x)|:x∈[0, n]}. Note that dX-convergence means L1-convergence plus uniform convergence on compacta. Theorem 2.4. The family Ais maximal dense-lineable. Proof. By Theorem 2.3 we already have the maximal lineability of the family A. It only remains to find an appropriate subset B⊂ C([0,+∞)) ∩ L1([0,+∞)) such that Bis dense-lineable and Ais stronger than B. Consider the family Bof all functions of the form bp,n,γ(x) =      p(x) if 0 ≤x≤n, p(n) γ(n+γ−x) if n<x≤n+γ, 0 if x>n+γ, where p(x) is a polygonal, n∈Nand γ > 0. Recall that a polygonal is a continuous function consisting of finitely many affine linear mappings on compact subintervals of [0,+∞). 7 It is obvious that B⊂ C([0,+∞))∩L1([0,+∞)) and that is a vector space. We are going to see that Bis dense in X. Let f∈ C([0,+∞)) ∩L1([0,+∞)) and ε > 0. Then there is N∈Nsuch that ∞ X k=N+1 1 2k<ε 6 and Z∞ N |f(x)|dx < ε 6. By using uniform continuity, it is easy to see that the set of all polygonals is dense in C([0, N]) even with the property that the approximating polygonal matches with the approximation function at the end of the interval. Consequently, we can take a polygonal p(x) in [0, N] such that p(N) = f(N) and kf−pk∞,N <ε 6N. Now define γ:= ε 6(1+|f(N)|). Then dX(f, bp,N,γ)≤ kf−pk∞,N N X k=1 1 2k+ ∞ X k=N+1 1 2k+ZN 0 |f(x)−p(x)|dx +ZN+γ Nf(x)−f(N) γ(N−x+γ)dx +Z∞ N+γ |f(x)|dx <ε 6·1 + ε 6+ε 6N·N+Z∞ N |f(x)|dx +|f(N)| γ·γ·γ+ε 6< ε. Hence, the set Ba dense vector space and, in particular, is dense-lineable. By its construction it is clear that any function in Bis bounded, so A∩B=∅ and A+B⊂ A. Finally, Lemma 2.2 gives us the maximal dense-lineability of A. In the following, we are going to establish the algebrability of A. But, before this, we need to introduce some notation. Given a (monic) monomial m(x1, . . . , xs) = Qs i=1 xαi i, where s∈Nand αi∈N∪ {0}, and the increasing sequence of prime numbers p= (pi)i, we define the p-index of mby indp(m) = m(p1, . . . , ps) = s Y i=1 pαi i. 8 Remark 2.5. Observe that the uniqueness of the factorization theorem states that, given a natural number n∈N, there is only one way of expressing it as product of powers of prime numbers. That is, there is an unique factorization n= s Y i=1 pαi i=p-index of a monomial m, which allows us to state a bijection between Nand the set of all monic monomials. That means that, given a monomial m, it is uniquely described by its p-index indp(m). Theorem 2.6. The family Ais strongly-algebrable. Proof. For any n∈N,p > 0, we consider the “triangles” on [0,+∞) given by Tn,p(x) =      np(2n+1x+ (1 −n2n+1))pif x∈n−1 2n+1 , n, np(−2n+1x+ (1 + n2n+1))pif x∈n, n +1 2n+1 , 0 otherwise, and we define the functions gp: [0,+∞)−→ Ras: gp(x) = ∞ X n=1 Tn,p(x). Because of the disjointness of the supports of the triangles Tn,p(x) (n∈N), the functions gpare well defined and continuous on [0,+∞). Furthermore, we can easily bound their L1-norm. Observe that the triangles Tn,p(x) are always “inside” the real triangles of basis n−1 2n+1 , n +1 2n+1 and height np (on x=n). Hence kgpkL1= ∞ X n=1 Z∞ 0 Tn,p(x)dx ≤ ∞ X n=1 np 2n<+∞. So, gp∈ C([0,+∞)) ∩L1([0,+∞)). Now, by evaluating gpon the sequence (xn)n= (n)nwe get that gp(xn) = gp(n) = npn→∞ −→ +∞, which yields gp∈ A for all p > 0. 9 Theorem 4.3. Let α: [0,+∞)−→ [1,+∞)be a continuous and nondecreasing function. The family A0(α)of sequences (fn)nof continuous and integrable functions with growth αsuch that fn→0in L1-norm and almost uniformly in [0,+∞)is maximal dense-lineable and strongly algebrable. 3. If we modify the construction of Example 2.1 by changing the triangles Tnto exponential-like functions, for instance Φn(x) := (n·e1−1 1−(2n+1(x−n))2if x∈n−1 2n+1 , n +1 2n+1  0 otherwise, we get for every n∈Nan integrable, unbounded and C∞function on [0,+∞). Hence a similar argument to the one given in Theorem 2.3 allows us to state the maximal lineability of A∞:= f∈ C∞([0,+∞)) ∩L1([0,+∞)) : lim sup x→+∞ |f(x)|= +∞. Moreover, because kΦnkL1≤1 2n, we also obtain the maximal lineability of the set A∞ 0of sequences (fn)nof functions in A∞such that fn→0 in L1- norm and almost uniformly in [0,∞). For the dense-lineability of both sets, we need first to state an adequate metric in Y:= C∞([0,+∞))∩L1([0,+∞)) and c0(Y); so we left it, together with the strong algebrability, as an open problem. Acknowledgment The authors would kindly acknowledge Prof. Bernal-Gonz´alez for his helpful and useful comments and suggestions. References [1] N. 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