# Injective mappings in [R.sup.R] and lineability.

1 Introduction and preliminaries

In  W. Goldbloom Bloch gives several examples of open mappings between Euclidean spaces which are discontinuous. One of the first examples of functions of this kind was proposed by Lebesgue, and is given as follows: let [DELTA] be the Cantor set and write R \ [DELTA] as a disjoint union of open sets {[I.sub.k]}. For every k, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As Goldbloom Bloch states, all the examples for an open discontinuous function require discontinuity at an infinite number of points. To give an example of a discontinuous open mapping which is continuous except at only a finite number of points, the author proposes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defined over C. This is an open mapping which is discontinuous only at 0. Using the usual identification C [left and right arrow] R2 we obtain an open mapping f : [R.sup.2] [right arrow] [R.sup.2] which is discontinuous only at (0, 0).

In the following, we will study the extent to which the number of points of discontinuity is important when trying to find algebraic structure in sets of discontinuous open mapping. Recall that in the beginning of the 21st century, the definitions that formalize the idea of "consider large algebraic structures inside of sets" were coined. To be more precise, we have the following:

Definition 1.1. [3, 4, 7, 10, 11] Let M be a subset of a topological vector space X and [micro] and n two cardinal numbers.

1. We say that M is [mu]-lineable if there exists a vector space V [subset or equal to] M [union] {0} of dimension [mu].

2. We say that M is [mu]-spaceable if there exists a closed vector space V [subset or equal to] M [union] {0} of dimension [mu]. We say that M is ([mu], n)-algebrable if there exists an algebra V [subset or equal to] M [union] {0} of dimension [mu] such that the cardinality of a minimal system of generators is n.

In the second half of the 19th century, the first examples for what we shall refer as (inmathematical folklore) pathological properties rose interest among mathematicians, and in 1875 Paul du Bois-Reymond published the proof for the existence of a continuous nowhere differentiable function. To express the shock that function provoked in most of mathematicians of that time, du Bois wrote: "It appears to me that the metaphysics of Weierstrass's function still hides many riddles and I cannot help thinking that entering deeper into thematter will finally lead us to the limits of our intellect."

The study of lineability (and other properties of subsets of topological vector spaces, together with the type of algebraic structure to be considered) tries to generalize the existence of those elements fulfilling pathological properties through finding large algebraic structures of such examples. Up to now, a great number of cases have been studied (giving even optimal results, talking about maximal dimension or cardinality). The monographs [2, 6] provide the state of the art of this topic.

In section 2, we will study two properties of functions: openness and, in relation with it, injectivity. It is interesting to stress that, even though most of the results that concerns lineability consist in finding explicitly structures of infinite dimension, we were able to prove that in most of this section that we can not go beyond finite dimension. One of the earliest results in this negative approach was the following:

Theorem 1.2. [1, 9, 16] The set of injective functions from R to R is not 2-lineable.

We will investigate the extent to which this theorem holds when seen in higher dimensions. We will use a corollary to the proof of the theorem, according to which, given f, g injective functions and x [not equal to] y, it is possible to construct a linear combination that attains the same value at x and y.

In the last section of this article, we will complete one problem that was presented in , where the authors studied smoothing properties of the Volterra convolution. Even though the definition slightly differs from the usual definition of convolution, this operator still takes the 'best properties' of the functions involved. Thus, for example, the Volterra convolution of an [L.sub.1]-function with a differentiable function having bounded derivative is still differentiable. Also, the Volterra convolution of two [L.sub.2]-functions gives a continuous function. Specifically, we have the following definition:

Definition 1.3. Let [f.sub.1], [f.sub.2] be two [L.sup.loc.sub.1](R)-functions. We define its Volterra convolution as

[f.sub.1] * V [f.sub.2](x) = [[integral].sup.x.sub.0] [f.sub.1](z) [f.sub.2](x - [zeta])d[zeta].

Observe that in the previous definition it is enough to consider locally [L.sub.1] functions, namely [L.sup.loc.sub.1](R).

The study of lineability or spaceability in relation to the smoothness of functions has been deeply carried out, and for example, in 1999, V.P. Font, V.I. Gurariy and M.I. Kadets proved the existence of a closed vector space of dimension c all of whose elements except zero are continuous nowhere differentiable functions ().

In , the authors were able to show that the set {[W.sub.a] : 7/9 < a < 1} is a linearly independent set which spans a vector space of such functions, where

[W.sub.a](x) = [summation over (k[greater than or equal to]0)] [a.sup.k] cos([9.sup.k]x)

is the well-known Weierstrass' Monster, and in , the authors proved that in fact any non-trivial linear combination of elements of this set gives a function whose convolution with itself is nowhere differentiable. With this result, the authors of  showed that the set of continuous nowhere differentiable functions whose convolution with themselves gives a nowhere differentiable function is c-lineable.

One might think that the property of nowhere differentiability would indeed imply the nowhere differentiability of the Volterra convolution. However, in  an example is given of a continuous nowhere differentiable function whose convolution with itself is everywhere differentiable. In section 3 we complement this last result by giving an independent set of cardinality c of nowhere differentiable functions such that, convolving two elements from the spanned vector space, we obtain a differentiable function.

2 Open, Continuous, and Injective mappings

As stated in the preliminaries, we will first separate the situation where the considered functions are discontinuous at an uncountable set of points or they are continuous except a finite number of points. For the first case, just remark that ES(R) := { f : R [right arrow] R : f (I) = R for every interval I} is a subset of the uncountably discontinuous open functions. We can then refer to , where it is proved that ES(R) is [2.sup.c]-lineable.

If we examine now the case where the open mappings are continuous except from a finite set, we have the following:

Theorem 2.1. The set of open mappings from R to R which are discontinuous at only a finite set of points is not 2-lineable.

Before giving the proof of this first result, we state a lemma that will be crucial for our purposes:

Lemma 2.2. A continuous open mapping f : R [right arrow] R is injective.

Proof of theorem 2.1. Assume [f.sub.1] and [f.sub.2] are two open mappings, so that [f.sub.1] is continuous on R \ {[x.sub.1],..., [x.sub.k]} and [f.sub.2] is continuous on R \ {[y.sub.1],..., [y.sub.l]}. Let us find (a, b) [subset or equal to] R so that [f.sub.1] and [f.sub.2] are continuous on (a, b).

We can choose then [a.sub.1], [a.sub.2] [member of] R \ {0} so that [a.sub.1] [f.sub.1] + [a.sub.2] [f.sub.2] is a nonzero noninjective continuous function on (a, b). Use then Lemma 2.2 to show that the restriction of [a.sub.1] [f.sub.1] + [a.sub.2] [f.sub.2] to (a, b) is not open (and hence [a.sub.1] [f.sub.1] + [a.sub.2] [f.sub.2] can not be open as a function on R).

Remark 2.3. An analogous proof also works if we study the functions that are continuous except on a discrete set, a subset of a closed set with finite Lebesgue measure or a bounded set.

The proof of Theorem 2.1 relies deeply in the fact that the set of injective functions defined from R to R is not 2-lineable (see Theorem 1.2 of Section 1). In the following result we will generalize Theorem 1.2 to arbitrary dimensions:

Theorem 2.4. Let n and m be arbitrary. Then, the set {f : [R.sup.n] [right arrow] [R.sup.m] : f is injective} is m-lineable, but not m + 1-lineable.

Proof. Let [PHI] : [R.sup.n] [right arrow] R be a bijection and consider the functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, {[f.sub.i] : 1 [less than or equal to] i [less than or equal to] m} is a linearly independent set, any nontrivial linear combination of whose elements gives an injective function. Indeed, let

f (x) = [m.summation over (i=1)] [a.sub.i] [f.sub.i](x).

and assume f(x) = f (y). Let [i.sub.0] = max{1 [less than or equal to] i [less than or equal to] m : [a.sub.i] [not equal to] 0}. Then, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so f (x) = f (y) yields [e.sup.[PHI](x)] = [e.sup.[PHI](y)] and the first part of the proof follows.

Let now [g.sub.1],..., [g.sub.m+1] : [R.sup.n] [right arrow] [R.sup.m] be functions and x [not equal to] y [member of] [R.sup.n]. Set, for 1 [less than or equal to] i [less than or equal to] m + 1, [g.sub.i](z) = ([g.sup.(1).sub.i](z),..., [g.sup.(m).sub.i] (z)) and define, for 1 [less than or equal to] k [less than or equal to] m,

[u.sub.k] = ([g.sup.(k).sub.1] (x) - [g.sup.(k).sub.1] (y),..., [g.sup.(k).sub.m+1](x) - [g.sup.(k).sub.m+1(y)]) [member of] [R.sup.m+1].

Then, dim_[u.sup.[perpendicular to].sub.1] [intersection] ... [intersection] [u.sup.[perpendicular to].sub.m]) [greater than or equal to] 1, so there exists b [member of] [R.sup.m+1] \ {0} such that <b, [u.sub.k]> = 0 for every 1 [less than or equal to] k [less than or equal to] m.

If we consider the function F = [b.sub.1][g.sub.1] +...+ [b.sub.m+1[g.sub.m+1]], we obtain that F cannot be injective. Indeed,

[F.sup.(k)](x) = [b.sub.1][g.sup.(k).sub.1] (x) + ... [b.sub.m+1][g.sup.(k).sub.m+1](x) = <b, [u.sub.k]> + [b.sub.1][g.sup.(k).sub.1](y) + ... [b.sub.m+1][g.sup.(k).sub.m+1](y) = [F.sup.(k)](y).

Related to this last result, we observe that the functions constructed in the proof are in general discontinuous (since we cannot have a continuous injective function [PHI] : [R.sup.n] [right arrow] R for n [greater than or equal to] 2). We can wonder if we can have a similar lineability result for continuous injective functions. Unfortunately, we have not been able to prove lineability for all dimensions, but we can at least give the following partial result for a subsequence of natural numbers:

Theorem 2.5. For every natural number n, the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is linear and bijective} is [2.sup.n]-lineable. In particular, the set of continuous injective functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [2.sup.n]-lineable.

The proof of this result relies in the following proposition, which we believe has its own interest:

Proposition 2.6. There exist [2.sup.n] symmetries [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the cube such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an orthogonal basis of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any x [not equal to] 0.

Proof. We will proceed by induction on n. Assume first n = 1. We may just choose

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we have identified [[PHI].sup.(1).sub.2] with the matrix defining a linear transformation. Assume next n = 2. In this case we may consider, similarly, the transformations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If x = (a, b, c, d), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and it is straightforward to see that the above is an orthogonal set.

Assume finally that for n [greater than or equal to] 2 the result holds true and let us prove the result for n + 1. Denote, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and define the operators

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constitutes an orthogonal system. For that, let us focus on the rows j and k, and let us denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that is, [sup.l][[PHI].sup.(n+1).sub.i] is the l-th component of [[PHI].sup.(n+1).sub.i] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] If first we assume 1 [less than or equal to] j, k [less than or equal to] [2.sup.n], then we need to compare the vectors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which are orthogonal (by induction hypothesis).

If we assume next [2.sup.n] + 1 [less than or equal to] j, k [less than or equal to] [2.sup.n+1], then by the same argument we also have orthogonality. Assume finally 1 [less than or equal to] j [less than or equal to] [2.sup.n] and [2.sup.n] + 1 [less than or equal to] k [less than or equal to] [2.sup.n+1]. Then, we need to compare the vectors

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and one can directly compute the inner product and show that is zero. This makes the induction step complete.

Proof of theorem 2.5. Consider the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by Proposition 2.6. Let

g(x) := [[2.sup.n].summation over (i=1)][[lambda].sub.i][[PHI].sub.i](x).

We will have the result if we succeed in proving that g is injective for any nontrivial choice of [[lambda].sub.i] [member of] R. Since g is linear, we must show that [g.sup.-1]({0}) = {0}. To this aim, let x [member of] [g.sup.-1]({0}). Then

[[2.sup.n].summation over (i=1)][[lambda].sub.i][[PHI].sub.i](x) = 0.

If we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then for each 1 [less than or equal to] j [less than or equal to] [2.sup.n] we obtain

[[2.sup.n].summation over (i=1)][[lambda].sub.i][[PHI].sup.j.sub.i] (x) = 0.

We can write this equality as the homogeneous system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By Proposition 2.6, the matrix of this system is non-degenerate if x is no zero. But the system has the non-trivial solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then x = 0 as required.

Remark 2.7. The proof of Theorem 2.5 is greatly supported by Proposition 2.6, but in fact we do not require the strong condition given there: it will be enough if, given a natural number m = [2.sup.n], we can guarantee the existence of linear operators [A.sub.2],..., [A.sub.m] : [R.sup.m] [right arrow] [R.sup.m] such that, for every x [member of] [R.sup.m] \ {0}, {x, [A.sub.2]x,..., [A.sub.m]x} is a linearly independent set. The last is somehow also a necessary condition for an arbitrary m [member of] N. Indeed, it is easy to see that the family of bijective linear functions [R.sup.m] [right arrow] [R.sup.m] is l-lineable if and only if there are bijective linear functions [f.sub.1],..., [f.sub.l] : [R.sup.m] [right arrow] [R.sup.m] such that, for every x [member of] [R.sup.m] \ {0}, the set {[f.sub.1](x),..., [f.sub.l](x)} is a linearly independent system.

Note that letting n = 2 in Theorem 2.5 we obtain that the family of bijective linear functions [R.sup.4] [right arrow] [R.sup.4] is 4-lineable. On the contrary, the corresponding assertion fails for [R.sup.3]:

Corollary 2.8. The set {f : [R.sup.3] [right arrow] [R.sup.3] : f is linear and bijective} is not 3-lineable.

Proof. Let us proceed by way of contradiction. According to Remark 2.7, we can assume, without loss of generality, that we can find non-singular linear operators A and B on [R.sup.3] so that, for every v = (x, y, z) [not equal to] 0, det[v, Av, Bv] [not equal to] 0, that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [e.sub.1] = (1, 0, 0), [e.sub.2] = (0, 1, 0) and [e.sub.3] = (0, 0, 1).

We are dealing then with a homogeneous polynomial, P(v = (x, y, z)) = det[v, Av, Bv], of degree 3, since in particular we have

det[[e.sub.1], [Ae.sub.1], [Be.sub.1]] * det[[e.sub.2], [Ae.sub.2], [Be.sub.2] x det[[e.sub.3], [Ae.sub.3], [B.sub.3]] [not equal to] 0.

If we focus our attention on the points of the form (x, 1, 0), we have that P(x, 1, 0) is a polynomial of one variable of degree 3, so in particular vanishes at one point [x.sub.0] [member of] R. Therefore, {([x.sub.0], 1, 0), A([x.sub.0], 1, 0), B([x.sub.0], 1, 0) cannot be a linearly independent set. This is the desired contradiction.

Remark 2.9. An analogous proof to the one in Corollary 2.8 would show that, for every natural number k, the set {f : [R.sup.2k+1] [right arrow] [R.sup.2k+1] : f is linear and bijective} is not (2k+1)-lineable.

Yet, we can have the partial positive result (from Theorem 2.5):

Corollary 2.10. For n [member of] N, write n = [2.sup.k]m, where 2 [??] m. Then, the set

{f : [R.sup.n] [right arrow] [R.sup.n] such that f is linear and bijective}

is [2.sup.k]-lineable. In particular, the set

{f : [R.sup.n] [right arrow] [R.sup.n] such that f is continuous and injective}

is 2k-lineable.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a basis giving the 2k-lineability of the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that f is linear and bijective .

Define [[bar.f].sub.i] : [R.sup.n] [right arrow] [R.sup.n], for 1 [less than or equal to] i [less than or equal to] [2.sup.k], as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] give the desired result.

All these considerations motivate the following open problem:

Question 2.11. What can we say about Theorem 2.5 for any dimension? Specifically, we are interested in the case when the dimension is even. Also, and taking into account what we have shown in Corollary 2.8, is {f : [R.sup.3] [right arrow] [R.sup.3] such that f is injective and continuous} (not) 3-lineable?

Continuing with the consequences of Theorem 2.5, we have the following result that deals with infinite dimensions:

Theorem 2.12. Let X be a real Banach space over R with Schauder basis [{[x.sub.j]}.sup.[infinity].sub.j=0] and let L(X) denote, as usual, the vector space of all operators (that is, the vector space of all continuous linear self-mappings). We have that {T [member of] L(X) such that T is injective} is lineable.

Proof. Assume that the Schauder basis of the Banach space X, [{[x.sub.j]}.sup.[infinity].sub.j=0], is normalized and let [V.sub.n] be a vector space provided by the [2.sup.n]-lineability (see Theorem 2.5) of the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that f is linear and injective}.

Let [{[f.sup.(n).sub.i]}.sup.2n.sub.i=1] be a basis of [V.sub.n].

Arrange then the elements of [[union].sub.n[member of]N][V.sub.n] in a matrix whose kth column consists of the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the first [2.sup.k-1] entries, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the other entries. That is, we create the following matrix:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let K [greater than or equal to] 1 be the basis constant associated to the Schauder basis [{[x.sub.j]}.sup.[infinity].sub.j=0]. Then, if x = [[SIGMA].sup.[infinity].sub.j=0] [a.sub.j][x.sub.j] is the unique representation of x [member of] X given by the Schauder basis [{[x.sub.j]}.sup.[infinity].sub.j=0], we have that [absolute value of [a.sub.j]] = [absolute value of [a.sub.j]] [parallel][x.sub.j][parallel] [less than or equal to] 2K[parallel]x[parallel] for all j [greater than or equal to] 0 (see e.g. [5, Chap. 2]).

Consider then the subset A = [{[F.sub.i]}.sub.i[member of]N] [subset] [[PI].sup.[infinity].sub.n=0] [V.sub.n] whose elements are the rows of the previous matrix. Notice that each [F.sub.i] has the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when-ever 2j [less than or equal to] n < [2.sup.j+1] (j = 0, 1, 2, ...). Next, define T([F.sub.i]) : X [right arrow] X by

T([F.sub.i])(x) = [[infinity].summation over (n=1)][c.sub.n][x.sub.n].

Now, by the selection of the functions [f.sup.(j).sub.i] (1 [less than or equal to] i [less than or equal to] [2.sup.j]) (see Theorem 2.5 and Proposition 2.6) we infer that the absolute value of each of the [2.sup.j] coordinates [b.sub.j,k] of the vector [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not bigger than [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and so not bigger than [2.sup.j] * 2K[parallel]x[parallel]. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which together with the completeness of X implies that [[SIGMA].sup.[infinity].sub.n=1] [c.sub.n][x.sub.n] converges in X. In other words, T([F.sub.i]) is well defined. Obviously, it is linear. Now, a simple application of the closed graph theorem (see e.g. [15, Chap. 2]) implies its continuity. Consequently, each T([F.sub.i]) [member of] L(X).

To conclude the result we shall show that every nontrivial linear combination of the set {T([F.sub.i]) : i [member of] N} yields an injective function. Thanks to linearity, this is equivalent to prove that the kernel of each such combination collapses to {0}.

Indeed, let f (x) = [[SIGMA].sup.n.sub.i=1] [[lambda].sub.i]T([F.sub.i])(x) and assume f ([[SIGMA].sup.[infinity].sub.j=0] [a.sub.j][x.sub.j]) = 0. Let k [greater than or equal to] 0 so that [2.sup.k] > n. Then, if we focus on the coordinates between [2.sup.k] and [2.sup.k+1] - 1 we would have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [{[lambda]i}.sup.n.sub.i=1] does not describe a trivial linear combination, we must conclude [[alpha].sub.i] = 0 for every 0 [less than or equal to] i [less than or equal to] [2.sup.k] - 1, due to the injectivity of [[lambda].sub.1] [f.sup.(k).sub.1] + ... + [[lambda].sub.n][f.sup.(k).sub.n]. Let k [right arrow] [infinity] to obtain the result.

We remark that the previous result cannot be improved for a general infinite dimensional topological space, in the following sense:

Theorem 2.13. The set {f : [c.sub.00] [right arrow] [c.sub.00] such that f is injective} is not [kappa]-lineable, for any cardinal number [kappa] > [N.sub.0].

Proof. Assume that we have a linearly independent set [{[f.sub.[alpha]]}.sub.[alpha][member of]G] of cardinality [kappa] > [N.sub.0]. Denote [f.sub.[alpha]] = ([f.sup.(1).sub.[alpha]], [f.sup.(2).sub.[alpha]], [f.sup.(1).sub.[alpha]], ...) and fix x, an element of [c.sub.00]. Then, there must exist a natural number n0 so that, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every n > [n.sub.0]}, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. By the same reason as before, there must exist a natural number m0 such that, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every m > [m.sub.0]}, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Take [k.sub.0] = max{[n.sub.0],[m.sub.0]} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (if [k.sub.0] = [n.sub.0]) or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (if [k.sub.0] = [m.sub.0]) and define, for every 1 [less than or equal to] i [less than or equal to] [k.sub.0] + 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore there must exist nonzero real numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so there exists a non trivial linear combination of the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is not injective.

Question 2.14. We would like to remark that c00 is not a Banach space. Therefore we have the following open question for this problem: Can we improve Theorem 2.12 for concrete spaces, like [c.sub.0]? Also, can we prove that the set there considered cannot be further lineable in general? Concerning the vector space c00, can we have an analogous theorem to Theorem 2.12 for this specific case?

3 Some remarks on nowhere differentiability

In , the authors used the functions [K.sub.a,b] defined by Knopp in some results in their article. By Z we denote, as usual, the set of all integer numbers.

Definition 3.1. Let 0 < a < 1 and b > 1 so that 1/a > ab > 1. Denote [f.sub.b,k](x) = dist([b.sup.k]x,Z) and define the Knopp's function as

[K.sub.a,b](x) = [summation over (k[greater than or equal to]0)] [a.sup.k][f.sub.b,k](x).

This function is a continuous nowhere differentiable function, but convolving it with a function of the same kind gives differentiability at every point.

Theorem 3.2. Let [K.sub.a] = [K.sub.a,9] for 1/9 < a < 1/3. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is differentiable every-where for all [a.sub.1], [a.sub.2] [member of] (1/9, 1/3).

In order to prove the above theorem, we first recall the following result, which follows from the well-known Weierstrass M-test:

Proposition 3.3. Let [([f.sub.n]).sup.[infinity].sub.n=0] be a sequence of differentiable functions on an interval I of R and let [([a.sub.n]).sup.[infinity].sub.n=0] be a sequence of numbers such that [[SIGMA].sup.[infinity].sub.n=0] [absolute value of [a.sub.n]] < [infinity]. Assume that [sup.sub.n[greater than or equal to]0] [sup.sub.x[member of]I] [absolute value of f'(x)] < [infinity] for all n [greater than or equal to] 0 and that [[SIGMA].sup.[infinity].sub.n=0] [a.sub.n][f.sub.n](x) converges for at least one x [member of] I. Then [[SIGMA].sup.[infinity].sub.n=0] [a.sub.n][f.sub.n] converges uniformly on I to a differentiable function f such that f' = [[SIGMA].sup.[infinity].sub.n=0] [a.sub.n][f'.sub.n].

Proof of Theorem 3.2. Denote [f.sub.k] = [f.sub.9,k] (see Definition 3.1) and notice

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with

[g.sub.k](x) = [(2/9[a.sub.1][a.sub.2] + 1).sup.k] [k.summation over (j=0)][a.sup.k.sub.1] [a.sub.2]k - j [[integral].sup.x.sub.0] dist([9.sup.k][tau])dist([9.sup.k-j](x - [tau]))d[tau],

where we have been able to interchange sum with integral sign using the fact that 0 < [a.sub.i] < 1, and therefore both infinite sums are convergent.

Each of the functions [g.sub.k] is differentiable, with

[g'.sub.k](x) = [(2/9[a.sub.1][a.sub.2] + 1).sup.k] [k.summation over (j=0)][a.sup.k.sub.1] [a.sup.k-j.sub.2] [[integral].sup.x.sub.0] dist([9.sup.k][tau])dist([9.sup.k-j](x - [tau]))d[tau],

and hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In conclusion, [absolute value of [g'.sub.k] (x)] [less than or equal to] 9M[a.sub.2]/2(1-9[a.sub.2]), for all x in [0,M] and all k. Applying Proposition 3.3 it follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is differentiable.

Corollary 3.4. The family {[K.sub.a] : 1/9 < a < 1/3} is a linearly independent set which spans a vector space of functions which, when convolving among themselves, give a differentiable function.

Proof. We only need to prove the linearly independence, just taking into account the linearity of the convolution and the differentiability. Indeed, let [[lambda].sub.1],..., [[lambda].sub.n] in R and 1/9 < [a.sub.1],..., [a.sub.n] < 1/3, and consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, assuming f [equivalent to] 0 and evaluating at 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

leading us to a Van der Monde determinant, which is nonzero because the [a.sub.i]'s are pairwise different. This allows us to conclude that [[lambda].sub.1] = 0 for every i, as desired.

Acknowledgements. The authors express their gratitude to the Referee and Prof. Richard Aron, whose thorough analysis and insightful remarks improved the text. The authors would also like to thank Prof. Sergii Myroshnychenko for pointing out the result gathered here in the form of Proposition 2.6, which was crucial for Section 2.

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* Supported by the Spanish Ministry of Science and Innovation, Grant MTM2015-65825-P.

Received by the editors in May 2016--In revised form in June 2016.

Communicated by F. Bastin.

2010 Mathematics Subject Classification : Primary 28[A.sub.2]0; Secondary 15A03, 26A24.

Department of Mathematical Sciences, Mathematics and Computer Science Building, Summit Street, Kent OH 44242, USA.

email:pjimen[e.sub.1]@kent.edu.

Department of Mathematics, University of Zanjan, Zanjan 45195-313, Iran.

email : s_maghso di@znu. ac. ir

Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain.

email:gustavo_fernandez@mat.ucm.es

Instituto de Ciencias Matematicas (CSIC-UAM-UC3M-UCM) C/ Nicolas Cabrera 13-15 Campus de Cantoblanco, 28049 Madrid, Spain

AND

Departamento de Analisis Matematico, Facultad de Ciencias Matematicas, Plaza de Ciencias 3, Universidad Complutense de Madrid, 28040 Madrid, Spain.

email:jseoane@ucm.es
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