Lineability in sequence and function spaces

LINEABILITY IN SEQUENCE AND FUNCTION SPACES G. ARA´ UJO*, L. BERNAL-GONZ´ ALEZ**, G.A. MU˜ NOZ-FERN´ ANDEZ***, J.A. PRADO-BASSAS**, AND J.B. SEOANE-SEP´ ULVEDA*** Abstract. It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors. 1. Introduction and notation Lebesgue ([25], 1904) was probably the first to show an example of a real function on the reals satisfying the rather surprising property that it takes on each real value in any nonempty open set (see also [20, 21]). The functions satisfying this property are called everywhere surjective (functions with even more stringent properties can be found in [16, 24]). Of course, such functions are nowhere continuous but, as we will see later, it is possible to construct a Lebesgue measurable everywhere surjective function. Entering a very different realm, in 1906 Pompeiu [27] was able to construct a nonconstant differentiable function on the reals whose derivative vanishes on a dense set. Passing to several variables, the first problem one meets related to the “minimal regularity” of functions at a elementary level is that of whether separate continuity implies continuity, the answer being given in the negative. In this paper, we will consider the families consisting of each of these kinds of functions, as well as two special families of sequences, and analyze the existence of large algebraic structures inside all these families. Nowadays the topic of lineability has had a major influence in many different areas on mathematics, from Real and Complex Analysis [9], to Set Theory 2010 Mathematics Subject Classification. Primary 28A20; Secondary 15A03, 26A24, 40A05, 46A45. Key words and phrases. Dense-lineability, algebrability, measurable function, differentiable function, separate continuity, divergent series. *Supported by PDSE/CAPES 8015/14-7. **Supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2012-34847-C02-01. ***Supported by the Spanish Ministry of Science and Innovation, grant MTM201234341. 1 arXiv:1507.04477v1 [math.FA] 16 Jul 2015 2 ARA´ UJO, BERNAL, MU˜ NOZ, PRADO, AND SEOANE 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 0 0.5 1 1.5 2 Figure 1. Rough sketch of the graph of Pompeiu’s original example. [19], Operator Theory [23], and even (more recently) in Probability Theory [15]. Our main goal here is to continue with this ongoing research. Let us now fix some notation. As usual, we denote by N,Qand Rthe set of positive integers, the set of rational numbers and the set of all real numbers, respectively. The symbol C(I) will stand for the vector space of all real continuous functions defined on an interval I⊂R. In the special case I=R, the space C(R) will be endowed with the topology of the convergence in compacta. It is well known that C(R) under this topology is an F-space, that is, a complete metrizable topological vector space. By MES it is denoted the family of Lebesgue measurable everywhere surjective functions R→R. A function f:R→Ris said to be a Pompeiu function (see Figure 1) provided that it is differentiable and f0vanishes on a dense set in R. The symbols Pand DP stand for the vector spaces of Pompeiu functions and of the derivatives of Pompeiu functions, respectively. Additional notation will be rather usual and, when needed, definitions will be provided. The organization of this paper is as follows. In section 2, a number of concepts concerning the linear or algebraic structure of sets inside a vector space or a linear algebra, together with some examples related to everywhere surjectivity and special derivatives, will be recalled. Sections 3, 4, and 5 will focus on diverse lineability properties of the families MES,P,DP, and certain subsets of discontinuous functions, so completing or extending a number of known results about several strange classes of real functions. Concerning sequence spaces, section 6 will deal with subsets of convergent and divergent series for which classical tests of convergence fail and, finally, in section 7 convergence in measure versus convergence almost everywhere will be analyzed in the space of sequences of measurable Lebesgue functions on the unit interval. LINEABILITY IN SEQUENCE AND FUNCTION SPACES 3 2. Lineability notions A number of concepts have been coined in order to describe the algebraic size of a given set; see [3, 5, 10, 14, 22] (see also the survey paper [12] and the forthcoming book [1] for an account of lineability properties of specific subsets of vector spaces). Namely, if Xis a vector space, αis a cardinal number and A⊂X, then Ais said to be: •lineable if there is an infinite dimensional vector space Msuch that M\ {0} ⊂ A, •α-lineable if there exists a vector space Mwith dim(M) = αand M\ {0} ⊂ A(hence lineability means ℵ0-lineability, where ℵ0= card (N), the cardinality of N), and •maximal lineable in Xif Ais dim (X)-lineable. If, in addition, Xis a topological vector space, then Ais said to be: •dense-lineable in Xwhenever there is a dense vector subspace Mof Xsatisfying M\{0} ⊂ A(hence dense-lineability implies lineability as soon as dim(X) = ∞), and •maximal dense-lineable in Xwhenever there is a dense vector subspace Mof Xsatisfying M\ {0} ⊂ Aand dim (M) = dim (X). And, according to [4,8], 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. •densely algebrable in Xif, in addition, Mcan be taken dense in X. •α-algebrable if there is an α-generated algebra Mwith M\{0} ⊂ A. •strongly α-algebrable if there exists an α-generated free algebra M with M\ {0} ⊂ A(for α=ℵ0, we simply say strongly algebrable). •densely strongly α-algebrable if, in addition, the free algebra Mcan be taken dense in X. Note that if Xis contained in a commutative algebra then a set B⊂X is a generating set of some free algebra contained in Aif and only if for any N∈N, any nonzero polynomial Pin Nvariables without constant term and any distinct f1, ..., fN∈B, we have P(f1, . . . , fN)6= 0 and P(f1, . . . , fN)∈ A. Observe that strong α-algebrability =⇒α-algebrability =⇒α-lineability, and none of these implications can be reversed; see [12, p. 74]. In [3] the authors proved that the set of everywhere surjective functions R→Ris 2c-lineable, which is the best possible result in terms of dimension (we have denoted by cthe cardinality of the continuum). In other words, the last set is maximal lineable in the space of all real functions. Other results establishing the degree of lineability of more stringent classes of functions can be found in [12] and the references contained in it. 4 ARA´ UJO, BERNAL, MU˜ NOZ, PRADO, AND SEOANE Turning to the setting of more regular functions, in [18] the following results are proved: the set of differentiable functions on Rwhose derivatives are discontinuous almost everywhere is c-lineable; given a non-void compact interval I⊂R, the family of differentiable functions whose derivatives are discontinuous almost everywhere on Iis dense-lineable in the space C(I), endowed with the supremum norm; and the class of differentiable functions on Rthat are monotone on no interval is c-lineable. Finally, recall that every bounded variation function on an interval I⊂R (that is, a function satisfying sup{Pn i=1 |f(ti)−f(ti−1)|:{t1< t2<· · · < tn} ⊂ I, n ∈N}<∞) is differentiable almost everywhere. A continuous bounded variation function f:I→Ris called strongly singular whenever f0(x) = 0 for almost every x∈Iand, in addition, fis nonconstant on any subinterval of I. Balcerzak et al. [6] showed that the set of strongly singular functions on [0,1] is densely strongly c-algebrable in C([0,1]). A number of results related to the above ones will be shown in the next two sections. 3. Measurable functions Our aim in this section is to study the lineability of the family of Lebesgue measurable functions R→Rthat are everywhere surjective, denoted MES. This result is quite surprising, since (as we can see in [17,18]), the class of everywhere surjective functions contains a 2c-lineable set of nonmeasurable ones (called Jones functions). Theorem 3.1. The set MES is c-lineable. Proof. Firstly, we consider the everywhere surjective function furnished in [18, Example 2.2]. For the sake of convenience, we reproduce here its construction. Let (In)n∈Nbe the collection of all open intervals with rational endpoints. The interval I1contains a Cantor type set, call it C1. Now, I2\C1also contains a Cantor type set, call it C2. Next, I3\(C1∪C2) contains, as well, a Cantor type set, C3. Inductively, we construct a family of pairwise disjoint Cantor type sets, (Cn)n∈N, such that for every n∈N, In\(Sn−1 k=1 Ck)⊃Cn. Now, for every n∈N, take any bijection φn:Cn→R, and define f:R→Ras f(x) = (φn(x) if x∈Cn, 0 otherwise. Then fis clearly everywhere surjective. Indeed, let Ibe any interval in R. There exists k∈Nsuch that Ik⊂I. Thus f(I)⊃f(Ik)⊃f(Ck) = φk(Ck) = R. But the novelty of the last function is that fis, in addition, zero almost everywhere, and in particular, it is (Lebesgue) measurable. That is, f∈ MES. LINEABILITY IN SEQUENCE AND FUNCTION SPACES 5 Now, taking advantage of the approach of [3, Proposition 4.2], we are going to construct a vector space that shall be useful later on. Let Λ := span {ϕα:α > 0}, where ϕα(x) := eαx−e−αx. Then Mis a c-dimensional vector space because the functions ϕα(α > 0) are linearly independent. Indeed, assume that there are scalars c1, . . . , cp(not all 0) as well as positive reals α1, . . . , αp such that c1ϕα1(x) + · · · +cpϕαp(x) = 0 for all x∈R. Without loss of generality, we may assume that p≥2, cp6= 0 and α1< α2<· · · < αp. Then lim x→+∞(c1ϕα1(x) + · · · +cpϕαp(x)) = +∞or −∞, which is clearly a contradiction. Therefore c1=· · · =cp= 0 and we are done. Note that each nonzero member g=Pp i=1 ciϕαi(with the ci’s and the αi’s as before) of Λ is (continuous and) surjective because limx→+∞g(x) = +∞and limx→−∞ g(x) = −∞ if cp>0 (with the values of the limits interchanged if cp<0). Next, we define the vector space M:= {g◦f:g∈Λ}. Observe that, since the fis measurable and the functions gin Λ are continuous, the members of Mare measurable. Fix any h∈M\ {0}. Then, again, there are finitely many scalars c1, . . . , cpwith cp6= 0, and positive reals α1< α2<· · · < αpsuch that g=c1ϕα1+· · · +cpϕαp and h=g◦f. Now, fix a non-degenerate interval J⊂R. Then h(J) = g(f(J)) = g(R) = R, which shows that his everywhere surjective. Hence M\ {0} ⊂ MES. Finally, by using the linear independence of the functions ϕαand the fact that fis surjective, it is easy to see that the functions ϕα◦f(α > 0) are linearly independent, which entails that Mhas dimension c, as required.  In [30, Example 2.34] it is exhibited one sequence of measurable everywhere surjective functions tending pointwise to zero. With Theorem 3.1 in hand, we now get a plethora of such sequences, and even in a much easier way than [30]. Corollary 3.2. The family of sequences {fn}n≥1of Lebesgue measurable functions R→Rsuch that fnconverges pointwise to zero and such that fn(I) = Rfor any positive integer nand each non-degenerate interval I, is c-lineable. Proof. Consider the family f Mconsisting of all sequences {hn}n≥1given by hn(x) = h(x)/n where the functions hrun over the vector space M constructed in the last theorem. It is easy to see that f Mis a c-dimensional vector subspace of (RR)N, that each hnis measurable, that hn(x)→0 (n→ ∞) for every x∈R, and that every hnis everywhere surjective if h is not the zero function.  6 ARA´ UJO, BERNAL, MU˜ NOZ, PRADO, AND SEOANE Remark 3.3. It would be interesting to know whether MES is –likewise the set of everywhere surjective functions– maximal lineable in RR(that is, 2c-lineable). 4. Special differentiable functions In this section, we analyze the lineability of the set of Pompeiu functions that are not constant on any interval. Of course, this set is not a vector space. Firstly, the following version of the well-known Stone–Weierstrass density theorem (see e.g. [28]) for the space C(R) will be relevant to the proof of our main result. Its proof is a simple application of the original Stone– Weierstrass theorem for C(S) (the Banach space of continuous functions S→R, endowed with the uniform distance, where Sis a compact topological space) together with the fact that convergence in C(R) means convergence on each compact subset of R. So we omit the proof. Lemma 4.1. Suppose that Ais a subalgebra of C(R)satisfying the following properties: (a) Given x0∈Rthere is F∈ A with F(x0)6= 0. (b) Given a pair of distinct points x0, x1∈R, there exists F∈ A such that F(x0)6=F(x1). Then Ais dense in C(R). In [6, Proposition 7], Balcerzak, Bartoszewicz and Filipczak established a nice algebrability result by using the so-called exponential-like functions, that is, the functions ϕ:R→Rof the form ϕ(x) = m X j=1 ajebjx for some m∈N, some a1, . . . , am∈R\ {0}and some distinct b1, . . . , bm∈ R\ {0}. By Ewe denote the class of exponential-like functions. The following lemma (see [11] or [1, Chapter 7]) is a slight variant of the mentioned Proposition 7 of [6]. Lemma 4.2. Let Ωbe a nonempty set and Fbe a family of functions Ω→R. Assume that there exists a function f∈ F such that f(Ω) is uncountable and ϕ◦f∈ F for every ϕ∈ E. Then Fis strongly c-algebrable. More precisely, if H⊂(0,∞)is a set with card(H) = cand linearly independent over the field Q, then {exp ◦(rf) : r∈H} is a free system of generators of an algebra contained in F ∪ {0}. Lemma 4.3 below is an adaptation of a result that is implicitly contained in [7, Section 6]. We sketch the proof for the sake of completeness. LINEABILITY IN SEQUENCE AND FUNCTION SPACES 7 Lemma 4.3. Let Fbe a family of functions in C(R). Assume that there exists a strictly monotone function f∈ F such that ϕ◦f∈ F for every exponential-like function ϕ. Then Fis densely strongly c-algebrable in C(R). Proof. If Ω = Rthen f(Ω) is a non-degenerate interval, so it is an uncountable set. Then, it is sufficient to show that the algebra Agenerated by the system {exp ◦(rf) : r∈H}given in Lemma 4.2 is dense. For this, we invoke Lemma 4.1. Take any α∈H⊂(0,+∞). Given x0∈R, the function F(x) := eα f(x)belongs to Aand satisfies F(x0)6= 0. Moreover, for prescribed distinct points x0, x1∈R, the same function Ffulfills F(x0)6=F(x1), because both functions fand x7→ eα x are one-to-one. As a conclusion, Ais dense in C(R).  Now we state and prove the main result of this section. Theorem 4.4. The set of functions in Pthat are nonconstant on any non-degenerated interval of Ris densely strongly c-algebrable in C(R). Proof. From [30, Example 3.11] (see also [29, Example 13.3]) we know that there exists a derivable strictly increasing real-valued function (a, b)→ (0,1) (with f((a, b)) = (0,1)) whose derivative vanishes on a dense set and yet does not vanish everywhere. By composition with the function x7→ b−a πarctan x+a+b 2, we get a strictly monotone function f:R→R satisfying that D:= {x∈R:f0(x)=0} is dense in Rbut D6=R. Observe that, in particular, fis a Pompeiu function that is nonconstant on any interval. According to Lemma 4.3, our only task is to prove that, given a prescribed function ϕ∈ E, the function ϕ◦fbelongs to F, where F:= {f∈ P :fis nonconstant on any interval of R}. By the chain rule, ϕ◦fis a differentiable function and (ϕ◦f)0(x) = ϕ0(f(x)) f0(x) (x∈R). Hence (ϕ◦f)0vanishes at least on D, so this derivative vanishes on a dense set. It remains to prove that ϕ◦fis nonconstant on any open interval of R. In order to see this, fix one such interval J. Clearly, the function ϕ0also belongs to E. Then ϕ0is a nonzero entire function. Therefore the set S:= {x∈R:ϕ0(x)=0} is discrete in R. In particular, it is closed in Rand countable, so R\S is open and dense in R. Of course, S∩(0,1) is discrete in (0,1). Since f:R→(0,1) is a homeomorphism, the set f−1(S) is discrete in R. Hence J\f−1(S) is a nonempty open set of J. On the other hand, since D is dense in R, it follows that the set D0of all interior points of Dis ∅. Indeed, if this were not true, there would exist an interval (c, d)⊂D. Then f0= 0 on (c, d), so fwould be constant on (c, d), which is not possible 8 ARA´ UJO, BERNAL, MU˜ NOZ, PRADO, AND SEOANE because fis strictly increasing. Therefore R\Dis dense in R, from which one derives that J\Dis dense in J. Thus (J\f−1(S)) ∩(J\D)6=∅. Finally, pick any point x0in the last set. This means that x0∈J,f(x0)6∈ S (so ϕ0(f(x0)) 6= 0) and x0/∈D(so f0(x0)6= 0). Thus (ϕ◦f)0(x0) = ϕ0(f(x0))f0(x0)6= 0, which implies that ϕ◦fis nonconstant on J, as required.  Remarks 4.5. 1. In view of the last theorem one might believe that the expression “f0= 0 on a dense set” (see the definition of P) could be replaced by the stronger one “f0= 0 almost everywhere”. But this is not possible because every differentiable function is an N-function –that is, it sends sets of null measure into sets of null measure– (see [29, Theorem 21.9]) and every continuous N-function on an interval whose derivative vanishes almost everywhere must be a constant (see [29, Theorem 21.10]). 2. If a real function fis a derivative then f2may be not a derivative (see [29, p. 86]). This leads us to conjecture that the set DP of Pompeiu derivatives (and of course, any subset of it) is not algebrable. 3. Nevertheless, from Theorem 3.6 (and also from Theorem 4.1) of [18] it follows that the family BDP of bounded Pompeiu derivatives is c-lineable. A quicker way to see this is by invoking the fact that BDP is a vector space that becomes a Banach space under the supremum norm [13, pp. 33–34]. Since it is not finite dimensional, a simple application of Baire’s category theorem yields dim (BDP) = c. Now, on one hand, we have that, trivially, BDP is dense-lineable in itself. On the other hand, it is known that the set of derivatives that are positive on a dense set and negative on another is a dense Gδset in the Banach space BDP [13, p. 34]. Then, as the authors of [18] suggest, it would be interesting to see whether this set is also denselineable. 5. Discontinuous functions Let n≥2 and consider the function f:Rn→Rgiven by (5.1) f(x1, . . . , xn) =        x1· · · xn x2n 1+· · · +x2n n if x2 1+· · · +x2 n6= 0, 0 if x1=· · · =xn= 0. Observe that fis discontinuous at the origin since arbitrarily near of 0 ∈Rn there exist points of the form x1=· · · =xn=tat which fhas the value 1 ntn. On the other hand, fixed (x1, . . . , xi−1, xi+1, . . . , xn)∈Rn−1, the real-valued function of a real variable given by ψ:xi7→ f(x1, . . . , xn) is everywhere a continuous function of xi. Indeed, this is trivial if all xj’s (j6=i) are not 0, while ψ≡0 if some xj= 0. Of course, fis continuous at any point of Rn\ {0}. LINEABILITY IN SEQUENCE AND FUNCTION SPACES 9 Given x0∈Rn, we denote by SC(Rn, x0) the vector space of all separately continuous functions Rn→Rthat are continuous on Rn\{x0}. Since card (C(Rn\ {x0})) = c, it is easy to see that the cardinality (so the dimension) of SC(Rn, x0) equals c. Theorem 5.1 below will show the algebrability of the family DSC(Rn, x0) := {f∈ SC(Rn, x0) : fis discontinuous at x0} in a maximal sense. Theorem 5.1. Let n∈Nwith n≥2, and let x0∈Rn. Then the set DSC(Rn, x0)is strongly c-algebrable. Proof. We can suppose without loss of generality that x0= 0 = (0,0,...,0). Consider the function f∈ DSC(Rn,0) given by (5.1). For each c > 0, we set ϕc(x) := e|x|c−e−|x|c. It is easy to see that these functions generate a free algebra. Indeed, if P(t1, . . . , tp) is a nonzero polynomial in pvariables with P(0,0,...,0) = 0 and c1, . . . , cpare distinct positive real numbers, let M:= {j∈ {1, . . . , p}: the variable tjappears explicitly in the expression of P}, and c0:= max{cj: j∈M}. Then one derives that the function P(ϕc1, . . . , ϕcp) has the form D em|x|c0+g(x)+h(x), where D∈R\ {0},m∈N,gis a finite sum of the form Pkmk|x|αkwith mkintegers and αk< c0, and his a finite linear combination of functions of the form eq(x)where, in turn, each q(x) is a finite sum of the form Pknk|x|γk, with each γksatisfying that either γk< c0, or γk=c0and nk<0 simultaneously. Then (5.2) lim x→∞ |P(ϕc1(x), . . . , ϕcp(x))|= +∞ and, in particular, P(ϕc1, . . . , ϕcp) is not 0 identically. This shows that the algebra Λ generated by the ϕc’s is free. Now, define the set Aas A={ϕ◦f:ϕ∈Λ}. Plainly, Ais an algebra of functions Rn→Reach of them being continuous on Rn\{0}. But, in addition, this algebra is freely generated by the functions ϕc◦f(c > 0). To see this, assume that Φ = P(ϕc1◦f, . . . , ϕcp◦f)∈ A, where P, c1, . . . , cpare as above. Suppose that Φ = 0. Evidently, the function fis onto (note that, for example, f(x, x, . . . , x) = 1 n xn,f(−x, x, x, . . . , x) =−1 n xnand f(0,...,0) = 0). Therefore P(ϕc1(x), . . . , ϕcp(x)) = 0 for all x∈R, so P≡0, which is absurd because P(ϕc1, . . . , ϕcp) becomes large as x→ ∞. Hence our only task is to prove that every function Φ ∈ A \ {0}as in the last paragraph belongs to DSC(Rn,0). Firstly, the continuity of each ϕc 16 ARA´ UJO, BERNAL, MU˜ NOZ, PRADO, AND SEOANE (where, for each n, the non-negative integers jand kare uniquely determined by n= 2k+jand 0 ≤j < 2k) satisfies that Tn→f≡0 in measure but, for every point x0∈[0,1], the sequence {Tn(x0)}n≥1does not converge. In order to face the lineability of this phenomenon, we need, once more, to put the problem in an adequate framework. Let LN 0be the space of all sequences of measurable functions [0,1] →R, endowed with the product topology. Since L0is metrizable and separable, the space LN 0is also a complete metrizable separable topological vector space. Again, by Baire’s theorem, this implies dim(LN 0) = c. Moreover, the set (7.1) Φ=(fn)∈LN 0:∃N=N(Φ) ∈Nsuch that fn= 0 for all n≥N is dense in the product space. Now, we are ready to state our next theorem, with which we finish this paper. Theorem 7.1. The family of Lebesgue classes of sequences (fn)∈LN 0such that fn→0in measure but (fn)does not converge almost everywhere in [0,1] is maximal dense-lineable in LN 0. Proof. Let (Tn) be the typewriter sequence defined above, and let Abe the family described in the statement of the theorem, so that (Tn)∈A. Extend each Tnto the whole Rby defining Tn(x) = 0 for all x /∈[0,1]. It is readily seen that, for each t∈(0,1/2), the translated-dilated sequence Tn,t(x) := Tn(2(x−t)) (n≥1) also tends to 0 in measure. Consider the vector space M:= span {(Tn,t) : t > 0}. The sequences (Tn,t) (0 <t<1/2) are linearly independent. Indeed, if it were not the case, there would be 0 < t1< t2<· · · < ts<1/2 as well as real numbers c1, . . . , cswith cs6= 0 such that c1Tn,t1+· · · +csTn,ts= 0 for all n∈N. In particular, c1T1,t1(x) + · · · +csT1,ts(x) = 0 for almost all x∈R. But T1=χ[0,1], so T1,t =χ[t,t+1 2]for all t > 0. Therefore c1χ[t1,t1+1 2](x) + · · · +csχ[ts,ts+1 2](x) = 0 for almost all x∈[0,1]. But, for every x∈(max{ts−1+1 2, ts}, ts+1 2], the left-hand side of the last expression equals 0+· · ·+0+cs·1 = cs6= 0, which is absurd. This shows the required linear independence. Then dim (M) = c. Moreover, since L0is a topological vector space carrying the topology of convergence in measure, we get that every member of (Fn) := (c1Tn,t1+· · · +csTn,ts)∈Mis a sequence tending to 0 in measure. Next, fix any (Fn)∈Mas above, with 0 < t1< t2<· · · < tsand cs6= 0. For all x∈(max{ts, ts−1+1 2}, ts+1 2], we have Fn(x) = s X j=1 cjTn,tj(x) = s X j=1 cjTn(2(x−tj)) = csTn(2(x−ts)). Since cs6= 0 and (Tn(y)) does not converge for every y∈(max{0,2(ts−1− ts)+1},1] (⊂[0,1]), we derive that, for each x∈(max{ts, ts−1+1 2}, ts+1 2], LINEABILITY IN SEQUENCE AND FUNCTION SPACES 17 the sequence (Fn(x)) does not converge. This shows that M\ {0} ⊂ A. Thus, Ais c-lineable. Finally, an application of Lemma 6.1 with X=LN 0and B= the set given by (7.1) puts an end on the proof.  References [1] R. M. Aron, L. Bernal-Gonz´alez, D. Pellegrino, and J. B. Seoane-Sep´ulveda, Lineability: The search for linearity in Mathematics, Monographs and Research Notes in Mathematics, Monographs and Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2015. [2] R. M. Aron, F. J. Garc´ıa-Pacheco, D. P´erez-Garc´ıa, and J. B. Seoane-Sep´ulveda, On dense-lineability of sets of functions on R, Topology 48 (2009), no. 2–4, 149–156. [3] R. M. Aron, V. I. Gurariy, and J. B. 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Rudin, Functional analysis, 2nd ed., McGraw-Hill Book Co., New York, 1991. [29] A. C. M. Van Rooij and W. H. Schikhof, A second course on real functions, Cambridge University Press, Cambridge, 1982. [30] G. L. Wise and E. B. Hall, Counterexamples in Probability and Real Analysis, Oxford University Press, USA, 1993. (G. Ara´ujo) Departamento de Matem´ atica, Universidade Federal da Para´ ıba, Jo˜ ao Pessoa - PB, 58.051-900 (Brazil) E-mail address:[email protected] (L. Bernal-Gonz´alez) Departamento de An´ alisis Matem´ atico, Universidad de Sevilla, Apdo. 1160, Avenida Reina Mercedes, Sevilla, 41080 (Spain). E-mail address:[email protected] (G. A. Mu˜noz-Fern´andez) Departamento de An´ alisis Matem´ atico, Facultad de Ciencias Matem´ aticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040 (Spain). E-mail address:gustavo [email protected] (J. A. Prado-Bassas) Departamento de An´ alisis Matem´ atico, Universidad de Sevilla, Apdo. 1160, Avenida Reina Mercedes, Sevilla, 41080 (Spain). E-mail address:[email protected] (J. B. Seoane-Sep´ulveda) ICMAT and Departamento de An´ alisis Matem´ atico, Facultad de Ciencias Matem´ aticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040 (Spain). E-mail address:[email protected]