Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities

Abstract

In this manuscript we characterize the completeness of a normed space through the strong lacunary (Nθ) and lacunary statistical convergence (Sθ) of series. A new characterization of weakly unconditionally Cauchy series throughNθandSθis obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesàro convergence summability space.

1. Introduction

Let X be a normed space, a sequence(_xk)⊂Xis said to be strongly 1-Cesàro summable (briefly,|_σ1|-summable) toL∈X if

limn→∞1n∑k=1n∥_xk−L∥=0.

Hardy-Littlewood [1] and Fekete [2] introduced this type of summability, which is related to the convergence of Fourier series (see [3,4]). The|_σ1| summability along with the statistical convergence [5] started a very striking theory with important applications [6,7,8]. Some years later, the strong lacunary summability_Nθ was presented by Freedman et al. [9] by introducing lacunary sequences and showed that_Nθis a larger class ofBK-spaces which had many of the characteristics of|_σ1| . Later on, Fridy [10,11] showed the concept of statistical lacunary summability and they related it with the statistical convergence and the_Nθsummability.

The characterization of a Banach space through different types of convergence has been dealt by authors such as Kolk [12], Connor, Ganichev, and Kadets [13],…

Consider X a normed space and∑_xi a series in X. In [14] the authors introduced the space of convergenceS(∑_xi)associated with the series∑_xi, which is defined as the space of sequences(_aj)in_ℓ∞such that∑_{ai xi}converges. They also proved that the necessary and sufficient condition for X to be a complete space is that for every weakly unconditionally Cauchy series∑_xi, the spaceS(∑_xi)is complete. Recall that∑_xiis a weakly unconditionally Cauchy (wuC) series if for every permutationπof the set of natural numbers, the sequence(_{∑i=1n xπ(i)})is a weakly Cauchy sequence. We will also rely on a powerful known result that states that a series∑_xiis wuC if and only if∑|f(_xi)|<∞for allf∈^X* (see [15] for Diestel’s complete monograph about series in Banach spaces).

In [16,17] a Banach space is characterized by means of the strong p-Cesàro summability (_wp) and ideal-convergence. In this manuscript, the_Nθand_Sθ summabilities are used along with the concept of weakly unconditionally series to characterize a Banach space. In Section 2 we introduce these two kinds of summabilities which are regular methods and we recall some properties. In Section 3 and Section 4 we introduce the spaces_SSθ (_{∑i xi})and_SNθ (_{∑i xi}) which will be used in Section 5 to characterize the completeness of a space.

2. Preliminaries

In this section, we present the definition of_Nθand_Sθsummabilities for Banach spaces and the relations between them. First, we recall the concept of lacunary sequences.

Definition 1.

A lacunary sequence is an increasing sequence of natural numbersθ=(_kr)such that_k0=0and_hr=_kr−_kr−1tends to infinite asr→∞. The intervals determined by θ will be denoted by_Ir=(_kr−1,_kr], the ratio_{kr kr−1}will be denoted by_qr.

We now give the definition of strong lacunary summability for Banach spaces based on the one given by Freedman for real-valued sequences [9].

Definition 2.

Let X be a Banach space andθ=(_kr)a lacunary sequence. A sequencex=(_xk)in X is lacunary strongly convergent or_Nθ−summabletoL∈Xiflimr→∞1_hr∑k∈_Ir∥_xk−L∥=0, and we write_Nθ-lim _xk=Lor_xk→_NθL.

Let_Nθbe the space of all lacunary strongly convergent sequences,

_Nθ=(_xk)⊆X:limr→∞1_hr∑k∈_Ir∥_xk−L∥=0forsomeL.

The space_Nθis a BK−space endowed with the norm∥_{xk ∥θ}=supr1_hr∑k∈_Ir∥_xk∥.

In 1993, Fridy and Orhan [11] introduced a generalization of the statistical convergence, the lacunary statistical convergence, using lacunary sequences. To accomplish this, they substituted the set{k:k≤n}by the set{k:_kr−1<k≤_kr}. We recall now the definition ofθ−density of a subsetK⊂N.

Definition 3.

Letθ=(_kr)be a lacunary sequence. IfK⊂N, theθ−density of K is denoted by_dθ(K)=limr1_hrcard({k∈_Ir:k∈K}),whenever this limit exists.

It is easy to show that this density is a finitely additive measure and we can define the concept of lacunary statistically convergent sequences for Banach spaces.

Definition 4.

Let X be a Banach space andθ=(_kr)a lacunary sequence. A sequencex=(_xk)is a lacunary statistically convergent sequence toL∈Xif givenε>0,

_dθ({k∈_Ir:∥_xk−L∥≥ε})=0,

or equivalently,

_dθ({k∈_Ir:∥_xk−L∥<ε})=1,

we say that(_xk)is_Sθ-convergent and we write_xk→_SθL.

Theorem 1.

Let X be a Banach space and(_xk)a sequence in X. Notice that_Sθand_Nθare regular methods.

Proof.

• If(_xk)→L, then(_xk)→_NθL.

  Letε>0, then there exists_k0such that ifk≥_k0, then

  ∥_xk−L∥<ε.

  Hence there exists_r0∈Nwith_r0≥_k0such that ifr≥_r0we have

  1_hr∑k∈_Ir∥_xk−L∥<1_hr∑k∈_Irε=_{hr hr}ε=ε

  which implies thatlimr→∞1_hr∑k∈_Ir∥_xk−L∥=0.

• If(_xk)→L, then(_xk)→_SθL.

  Simply observe that since(_xk)→L, givenε>0there exists_k0such that for everyk≥_k0we getcard({k∈_Ir:∥_xk−L∥≥ε})=0, which implies_dθ({k∈_Ir:∥_xk−L∥≥ε})=0for everyk≥_k0. □

The reverse is not true, as we will show in Example 1, in which we introduce an unbounded sequence that is_Nθ-summable and Example 2 where an unbounded_Sθconvergent sequence is presented.

Example 1.

There exist unbounded sequences which are_Nθ-summable.

Letθ=(_kr)be the lacunary sequence with_k0=0and_kr=^2r. Notice that

• _h1=_k1−_k0=2and_hr=^2r−1for everyr≥2.

• _I1=(_k0,_k1]=(0,2]and_Ir=(^2r−1,^2r]for everyr≥2.

Consider the sequence defined by

_xk=0ifk≠^2jforallj,j−1ifk=^2jforsomej.

Notice that(_xk)is unbounded and observe that

∑k∈_Ir|_xk−0|_hr=0ifr=1,r−1^2r−1ifr≥2,⟶r→∞0,

which implies that_xk→_Nθ0.

Fridy and Orhan [10] showed that_Nθand_Sθare equivalent for real-valued bounded sequences. This fact also holds for Banach spaces and we include the proof for the sake of completeness.

Theorem 2.

Let X be a Banach space,(_xk)a sequence in X andθ=(_kr)a lacunary sequence. Then:

1.

(_xk)→_NθLimplies(_xk)→_SθL.

2.

(_xk)bounded and(_xk)→_SθLimply(_xk)→_NθL.

Proof.

1. If(_xk)→_NθL, then for everyε>0,

∑k∈_Ir∥_xk−L∥≥∑k∈_Ir∥_xk−L∥≥ε∥_xk−L∥≥εcard({k∈_Ir:∥_xk−L∥≥ε}),

which implies that(_xk)→_SθL.

2. Let us suppose that(_xk)is bounded and(_xk)→_SθL. Since(_xk)is bounded, there existsM>0such that∥_xk−L∥≤Mfor everyk∈N. Givenε>0,

1_hr∑k∈_Ir∥_xk−L∥=1_hr∑k∈_Ir∥_xk−L∥≥ε∥_xk−L∥+1_hr∑k∈_Ir∥_xk−L∥<ε∥_xk−L∥≤M_hrcard({k∈_Ir:∥_xk−L∥≥ε})+ε,

so we deduce that(_xk)→_NθL. □

Next, we give an example to illustrate that the hypothesis over the sequence to be bounded is necessary and cannot be removed.

Example 2.

There exist unbounded_Sθ-convergent sequences to L which are not_Nθ-summable to L.

Letθ=(_kr)be the lacunary sequence with_k0=0and_kr=^2r. Notice that

• _h1=_k1−_k0=2and_hr=^2r−1for everyr≥2.

• _I1=(_k0,_k1]=(0,2]and_Ir=(^2r−1,^2r]for everyr≥2.

Consider the sequence defined by

_xk=0ifk≠^2jforallj,^2jifk=^2jforsomej.

Givenε>0, it is easy to show thatcard({k∈_Ir:|_xk−0|≥ε})_hr→0asr→∞, which implies that(_xk)→_Sθ0. Also, notice that(_xk)is an unbounded sequence. However,

∑k∈_Ir|_xk−0|_hr=22=1ifr=1,^{2r 2r−1}=2ifr≥2,⟶r→∞2,

which implies that_xk↛_Nθ0.

We now give the definition of lacunary statistically Cauchy sequences in Banach spaces as a generalization of the definition for real-valued sequences by Fridy and Orhan in [11].

Definition 5.

Let X be a Banach space andθ=(_kr)a lacunary sequence. A sequencex=(_xk)is a lacunary statistically Cauchy sequence if there exists a subsequence_x^k′(r)of_xksuch that^k′(r)∈_Irfor everyr∈N,limr→∞_x^k′(r)=Lfor someL∈Xand for everyε>0,

limr→∞1_hrcard({k∈_Ir:∥_xk−_x^k′(r)∥≥ε})=0,

or equivalently,

limr→∞1_hrcard({k∈_Ir:∥_xk−_x^k′(r)∥<ε})=1.

In this case, we say that(_xk)is_Sθ-Cauchy.

An important result in [11] is the_Sθ-Cauchy Criterion and some of the next theorems in this work rely on it. This result can also be obtained for sequences in Banach spaces, and we include the proof for the sake of completeness.

Theorem 3.

Let X be a Banach space. A sequence(_xk)in X is_Sθ-convergent if and only if it is_Sθ-Cauchy.

Proof.

Let(_xk)be an_Sθ-convergent sequence in X and for everyk∈N, we define_Kj={k∈N:∥_xk−L∥<1/j}. Observe that_Kj⊇_Kj+1andcard(_Kj∩_Ir)_hr→1asr→∞.

Set_m1such that ifr≤_m1thencard(_K1∩_Ir)/_hr>0, i.e.,_K1∩_Ir≠⌀. Next, choose_m2>_m1such that ifr≥_m2, then_K2∩_Ir≠⌀. Now, for each_m1≤r≤_m2, we choose_kr′∈_Irsuch that_kr′∈_Ir∩_K1, i.e.,∥_xkr′ −L∥<1. Inductively, we choose_mp+1>_mpsuch that ifr>_mp+1, then_Ir∩_Kp+1≠⌀. Thus, for all r such that_mp≤r<_mp+1, we choose_kr′∈_Ir∩_Kp, and we have∥_xkr′ −L∥<1/p.

Therefore, we have a sequence_kr′such that_kr′∈_Irfor everyr∈Nand_{limr→∞ xkr′} =L. Finally,

1_hrcard({k∈_Ir:∥_xk−_xkr′ ∥≥ε})≤1_hrcard({k∈_Ir:∥_xk−L∥≥ε/2})+1_hrcard({k∈_Ir:∥_xkr′ −L∥≥ε/2}).

Since(_xk)→_SθLand_{limr→∞ xkr′} =Lwe deduce that(_xk)is_Sθ- Cauchy.

Conversely, if(_xk)is a Cauchy sequence, for everyε>0,

card({k∈_Ir:∥_xk−L∥}∥≥ε})≤card({k∈_Ir:∥_xk−_xkr′ ∥≥ε/2})+card({k∈_Ir:∥_xkr′ −L∥≥ε/2}).

Since(_xk)is_Sθ-Cauchy and_{limr→∞ xkr′} =L, we deduce that(_xk)→_SθL. □

3. The Statistical Lacunary Summability Space

Let us consider X a real Banach space,_{∑i xi}a series in X andθ=(_kr)a lacunary sequence. We define

_SSθ ∑i_xi=_(ai)i∈_ℓ∞:∑i_{ai xi} is _Sθ-summable

endowed with the supremum norm. This space will be named as the space of_Sθ-summability associated with_{∑i xi}. We will characterize the completeness of the space_{SSθ ∑i xi}in Theorem 4, but first we need a lemma.

Lemma 1.

Let X be a real Banach space and suppose that the series∑_xiis not wuC. Then there existf∈^X*and a null sequence_(ai)i∈_c0such that

∑i_aif(_xi)=+∞

and

_aif(_xi)≥0.

Proof.

Since_∑i=1∞|f(_xi)|=+∞, there exists_m1such that_∑i=1m1 |f(_xi)|>2·2.

We define_ai=12iff(_xi)≥0and_ai=−12iff(_xi)<0fori∈{1,2,⋯,_m1}.

This implies that_{∑i=1m1 ai}f(_xi)>2and_aif(_xi)≥0ifi∈{1,2,⋯,_m1}.

Let_m2>_m1be such that_∑i=m1+1m2 |f(_xi)|>²²·²².

We define_ai=1²²iff(_xi)≥0and_ai=−1²²iff(_xi)<0fori∈{_m1+1,⋯,_m2}. Hence_{∑i=m1+1m2 ai}f(_xi)>²²and_aif(_xi)≥0ifi∈{_m1+1,⋯,_m2}. So we have obtained a sequence_(ai)i∈_c0with the above properties. □

Theorem 4.

Let X be a real Banach space andθ=(_kr)a lacunary sequence. The following are equivalent:

(1)

The series_{∑i xi}is weakly unconditionally Cauchy (wuC).

(2)

The space_SSθ (_{∑i xi})is complete.

(3)

The space of all null sequences_c0is contained in_SSθ (_{∑i xi}).

Proof.

(1)⇒(2): Since∑_xiis wuC, the following supremum is finite:

H=sup∑i=1n_{ai xi}:|_ai|≤1,1≤i≤n,n∈N<+∞.

Let_(^am)m⊂_SSθ (_{∑i xi})such thatlimm_{∥^am−^a0∥∞}=0, with^a0∈_ℓ∞. We will show that^a0∈_SSθ (_{∑i xi}). Let us suppose without any loss of generality that∥^a0 _∥∞≤1. Then, the partial sums_Sk0=_{∑i=1k ai0 xi}satisfy∥_Sk0∥≤Hfor everyk∈N, i.e., the sequence(_Sk0)is bounded. Then,^a0∈_SSθ (_{∑i xi})if and only if(_Sk0)is_Sθ-summable to someL∈X. According to Theorem 3,(_Sk0)is lacunary statistically convergent toL∈Xif and only if(_Sk0)is a lacunary statistically Cauchy sequence.

Givenε>0andn∈N, we obtain statement (2) if we show that there exists a sub-sequence(_S^k′(r))such that^k′(r)∈_Irfor every r,limr→∞_S^k′(r)=Land

_dθ({k∈_Ir:∥_Sk0−_S^k′(r)0∥<ε})=1.

Since^am→^a0in_ℓ∞, there exists_m0>nsuch that∥^am−^a0 _∥∞<ε4Hfor allm>_m0, and since_Skm0 is_Sθ-Cauchy, there exists^k′(r)∈_Irsuch thatlimr→∞_S^k′(r)m0 =Lfor some L and

_dθk∈_Ir:∥_Skm0 −_S^k′(r)m0 ∥<ε2=1.

Considerr∈Nand fixk∈_Irsuch that

∥_Skm0 −_S^k′(r)m0 ∥<ε2.

We will show that∥_Sk0−_S^k′(r)0∥<ε, and this will prove that

k∈_Ir:∥_Skm0 −_S^k′(r)m0 ∥<ε2⊂{k∈_Ir:∥_Sk0−_S^k′(r)0∥<ε}.

Since the first set has density 1, the second will also have density 1 and we will be done.

Let us observe first that for everyj∈N,

∑i=1j4Hε(_aim−_aim0 )_xi≤H,

for everym>_m0, therefore

_Sj0−_Sjm0 =∑i=1j(_ai0−_aim0 )_xi≤ε4.

Then, by applying the triangular inequality,

∥_Sk0−_S^k′(r)0∥≤∥_Sk0−_Skm0 ∥+∥_Skm0 −_S^k′(r)m0 ∥+∥_S^k′(r)m0 −_S^k′(r)0∥<ε4+ε2+ε4=ε.

where the last inequality follows by applying (1) and (2), which yields the desired result.

(2)⇒(3): Let us observe that if_SSθ (_{∑i xi})is complete, then it contains the space of eventually zero sequences_c00and therefore the thesis comes, since the supremum norm completion of_c00is_c0.

(3)⇒(1): By way of contradiction, suppose that the series∑_xiis not wuC. Therefore there existsf∈^X*such that∑i=1∞|f(_xi)|=+∞. By Lemma 1 we can construct inductively a sequence_(ai)i∈_c0such that

∑i_aif(_xi)=+∞

and

_aif(_xi)≥0.

Now we will prove that the sequence_Sk=_{∑i=1k ai}f(_xi)is not_Sθ-summable to anyL∈R. By way of contradiction, suppose that it is_Sθ-summable toL∈R, then we have

1_hrcard({k∈_Ir:|_Sk−L|≥ε})=1_hr∑k=_kr−1|_Sk−L|≥ε_kr1→r→∞0.

Since_Skis an increasing sequence and_Sk→∞, there exists_k0such that|_Sk−L|≥εfor everyk≥_k0. Let us suppose that_kr>_k0for every r. Hence,

1_hr∑k=_kr−1|_Sk−L|≥ε_kr1=_{hr hr}=1↛r→∞0,

which is a contradiction. This implies that_Skis not_Sθ-convergent and this is a contradiction with (3). □

4. The Strong Lacunary Summability Space

Let X be a real Banach space,_{∑i xi}a series in X andθ=(_kr)a lacunary sequence. We define

_SNθ ∑i_xi=_(ai)i∈_ℓ∞:∑i_{ai xi}is_Nθ-summable

endowed with the supremum norm. This will be named as the space of_Nθ-summability associated with the series_{∑i xi}. We can now present a theorem very similar to that of Theorem 4 but for the case of_Nθ-summability. Indeed Theorem 5 characterizes the completeness of the space_{SNθ ∑i xi}.

Theorem 5.

Let X be a real Banach space andθ=(_kr)a lacunary sequence. The following are equivalent:

(1)

The series_{∑i xi}is weakly unconditionally Cauchy (wuC).

(2)

The space_SNθ (_{∑i xi})is complete.

(3)

The space of all null sequences_c0is contained in_SNθ (_{∑i xi}).

Proof.

(1) ⇒ (2): Since∑_xiis wuC, the following supremum is finite

H=sup∑i=1n_{ai xi}:|_ai|≤1,1≤i≤n,n∈N<+∞.

Let_(^am)m⊂_SNθ (_{∑i xi})such thatlimm_{∥^am−^a0∥∞}=0, with^a0∈_ℓ∞.

We will show that^a0∈_SNθ (_{∑i xi}).

Without loss of generality we can suppose that∥^a0 _∥∞≤1. Therefore the partial sums_Sk0=_{∑i=1k ai0 xi}satisfy∥_Sk0∥≤Hfor everyk∈N, i.e., the sequence(_Sk0)is bounded. Hence^a0∈_SNθ (_{∑i xi})if and only if(_Sk0)is_Nθ-summable to someL∈X. Since(_Sk0)is bounded, it is sufficient to show that(_Sk)is_Sθ -convergent, thanks to to Fridy and Orhan’s Theorem ([10], Theorem 2.1) (see Theorem 2). The result follows analogously as in Theorem 4.

(2)⇒(3): It is sufficient to notice that_SSθ (_{∑i xi})is a complete space and it contains the space of eventually zero sequences_c00, so it contains the completion of_c00with respect to the supremum norm, hence it contains_c0.

(3)⇒(1): By way of contradiction, suppose that the series∑_xiis not wuC. Therefore there existsf∈^X*such that∑i=1∞|f(_xi)|=+∞. By Lemma 1 we can construct inductively a sequence_(ai)i∈_c0such that_{∑i ai}f(_xi)=+∞and_aif(_xi)≥0.

The sequence_Sk=_{∑i=1k ai}f(_xi)is not_Nθ-summable to anyL∈R.

As_Sk→∞, for everyA>0, there exists_k0such that|_Sk|>Aifk≥_k0. Then we have

1_hr∑k∈_Ir|_Sk|>_hrA_hr=A.

Hence_Skis not_Nθ-summable to anyL∈R, otherwise

∞←1_hr∑k∈_Ir|_Sk|≤|L|+1_hr∑k∈_Ir|_Sk−L|→|L|

We can conclude that_Skis not_Nθ-convergent, a contradiction with (3). □

5. Characterizations of the Completeness of a Banach Space

A Banach space X can be characterized by the completeness of the space_SNθ (_{∑k xk})for every wuC series_{∑k xk}, as we will show next.

Theorem 6.

Let X be a normed real vector space. Then X is complete if and only if_SNθ (_{∑k xk})is a complete space for every weakly unconditionally Cauchy series (wuC)_{∑k xk}.

Proof. Thanks to Theorem 4, the condition is necessary.

Now suppose that X is not complete, hence there exists a series∑_xkin X such that∥_xk∥≤1k^2kand∑_xk=^x**∈^X**\X.

We will provide a wuC series_{∑k yk}such that_SNθ (_{∑k yk})is not complete, a contradiction.

Set_SN=∑k=1N_xk. As^X**is a Banach space endowed with the dual topology,sup∥^y*∥≤1|^y*(_SN)−^x**(^y*)|tends to 0 asN→∞, i.e.,

limN→+∞^y*(_SN)=limN→+∞∑k=1N^y*(_xk)=^x**(^y*),forevery∥^y*∥≤1.

Put_yk=k_xkand let us observe that∥_yk∥<1^2k. Therefore∑_ykis absolutely convergent, thus it is unconditionally convergent and weakly unconditionally Cauchy.

We claim that the series∑k1k_ykis not_Nθ-summable in X.

By way of contradiction suppose that_SN=_∑k=1N1k_ykis_Nθ-summable in X, i.e., there exists L in X such thatlimr→∞1_hr∑k∈_Ir∥_Sk−L∥=0. This implies that

limr→+∞1_hr∑k∈_Ir^y*(_Sk)=^y*(L),forevery∥^y*∥≤1.

From Equations (3) and (4), the uniqueness of the limit and since_Nθis a regular method, we have^x**(^y*)=^y*(L)for every∥^y*∥≤1, so we obtain^x**=L∈X, a contradiction. Hence_SN=_∑k=1N1k_ykis not_Nθ-summable to anyL∈X.

Finally, let us observe that since_{∑k yk}is a weakly unconditionally Cauchy series and_SN=_∑k=1N1k_ykis not_Nθ-summable, we have(1k)∉_SNθ (_{∑k yk})and this means that_c0⊈_SNθ (_{∑k yk})which is a contradiction with Theorem 5(3), so the proof is complete. □

By a similar argument and taking into account Theorem 2, we have also the characterization for the_Sθ-summability:

Theorem 7.

Let X be a normed real vector space. Then X is complete if and only if_SSθ (_{∑i xi})is a complete space for every weakly unconditionally Cauchy series (wuC)_{∑i xi}.

Let0<p<+∞, the sequence(_xn)is said to be strongly p-Cesàro or_wp-summable if there isL∈Xsuch that

limn1n∑i=1n^{∥_xi−L∥p}=0;

in this case we will write(_xk)→_wpLandL=_wp−_{limn xn}.Let∑_xibe a series in a real Banach space X, let us define

_Swp ∑i_xi=_(ai)i∈_ℓ∞:∑i_{ai xi}is_wp-summable

endowed with the supremum norm. We refer to [16] for other properties of the space_Swp (_{∑i xi}).

Finally, from Theorem 6, Theorem 7 and ([16], Theorem 3.5), we derive the following corollary.

Corollary 1.

Let X be a normed real vector space andp≥1. The following are equivalent:

1. X is complete.

2.

_SNθ (_{∑k xk})is complete for every weakly unconditionally Cauchy series (wuC)_{∑k xk}.

3.

_SSθ (_{∑k xk})is complete for every weakly unconditionally Cauchy series (wuC)_{∑k xk}.

4.

_Swp (_{∑k xk})is complete for every weakly unconditionally Cauchy series (wuC)_{∑k xk}.

Author Contributions

Conceptualization, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; methodology, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; formal analysis, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; investigation, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; writing-original draft preparation, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; writing-review and editing, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; visualization, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; supervision, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; project administration, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P.; funding acquisition, S.M.-P., G.B., F.L.-S., F.J.P.-F. and A.S.-P. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported by the FQM-257 research group of the University of Cádiz and the Research Grant PGC-101514-B-100 awarded by the Spanish Ministry of Science, Innovation and Universities and partially funded by the European Regional Development Fund. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Acknowledgments

The authors would like to thank the reviewers for valuable comments that helped improve the manuscript considerably.

Conflicts of Interest

The authors declare no conflict of interest.

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