EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 3, 2023, 1762-1771 ISSN 1307-5543 – ejpam.com Published by New York Business Global Nuclearity of a class of vector-valued sequence spaces Mohamed Ahmed Ould Sidaty1,2 1 Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, Kingdom of Saudi Arabia 2 École Normale Supérieure de Nouakchott, Mauritanie Abstract. In this note, we deal with a perfect sequence space λ and a convex bornological space E to introduce and study the space λ(E) of all totally λ-summable sequences from E. We prove that λ(E) is complete if and only if λ and E are complete, nuclear if and only if λ and E are nuclear, and we make use of a result of Ronald C. Rosier [10] to give a similar characterization of the nuclearity of the space λ{E} of all absolutely λ−summable sequences in a locally convex E. 2020 Mathematics Subject Classifications: 46A17, 46A45, 47B37, 46B45 Key Words and Phrases: Sequence spaces, convex bornological spaces, locally convex sequence spaces, nuclearity, summability Introduction In connection with the nuclearity of a locally convex space E, A. Pietsch in [9] in- troduced the spaces ℓp(E) and ℓp{E} respectively of weakly ℓp-summable and absolutely ℓp-summable sequences in E. In [8], he used these spaces to study the absolutely p-summing operators. Later, he introduced and studied also the space λ{E} of λ-summable sequences in E, for a perfect sequence space λ in the sense of Köthe endowed with its normal topol- ogy. Many other authors were interested in the study of these spaces. Ronald C. Rosier in [10] considered a general polar topology on λ{E} and got a precise description of the topological dual and its equicontinuous subsets. M. Florencio and P. J. Paúl [3], con- sidering general polar topologies, obtained many interesting results such as barreledness conditions. In [1] and [2], they studied the space λ(E) of weakly λ−summables sequences in E and represented this space as the completion of the injective tensor product λ⊗̃ϵE. In [6] and [7], L. Oubbi and M. A. Ould Sidaty reconsidered the space λ(E) and obtained some of its properties. They mainly described the continuous dual space of λ(E). While in [11] and [13], characterizations of the reflexivity of λ(E) in terms of that of λ and E and the AK-property are given. A characterization of the nuclearity of of the space of weakly λ−summable sequences is given in [12]. DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4831 Email address: sidaty1@hotmail.com (M. A. Sidaty) https://www.ejpam.com 1762 © 2023 EJPAM All rights reserved. M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1763 In this note, we are concerned with the nuclearity of the convex bornological space λ(E) of all totally λ−summable sequences in E, in the sense of [3], where E is a convex bornolog- ical space. In sections 1 and 2, we endow this space with a structure of b-space, and study some of its properties. The section 3 is devoted to the nuclearity of λ(E). We prove mainly that λ(E) possesses this property if and only if both of λ and E have. In Section 4, we provide an application of the results of Section 3 on the nuclearity of the space λ{E} of absolutely λ−summable sequences in a locally convex space E. 1. Preliminaries For a linear space E, we mean by a convex bornology on E, a collection of subsets of E covering E, hereditary for the inclusion, and closed for the finite unions, the addition, the scalar multiplication and the formation of absolutely convex hulls. We say then that E is a convex bornological space or simply a b-space. The elements of the bornology of E are called bounded sets of E. A collection B of bounded sets of E is a basis for its bornology if every bounded set in E is contained in an element of B. In the sequel, we assume that the members of B are absolutely convex. A b-space E is said to be Hausdorff if the only bounded linear subspace of E is {0}. We say that a sequence {xn}∞n=1 ⊂ E converges to x ∈ E, or that x is a limit of {xn}∞n=1 in E if there exists an element B ∈ B such that {xn − x}∞n=1 is contained and convergent to 0 in the normed space (EB, ∥ · ∥B), where EB is the subspace of E generated by B and ∥ · ∥B is the gauge of B. A subset of a b-space E will be said to be closed if it contains the limits of all its sequences. A Banach disk in a b-space E is an element B ∈ B for which the normed space EB is complete. E is said to be b-complete or simply complete if every bounded set in E is contained in a Banach disk in E. A linear mapping between two b-spaces E and F is said to be bounded if it transforms bounded sets of E to bounded sets of F . A bounded linear mapping transforms convergent sequences to convergent ones. A bornological isomorphism is a bounded linear bijection whose inverse is also bounded. The Köthe dual of a sequence space λ is defined as λ× = { (βn) ⊂ C : ∞∑ n=1 |αnβn| converges for all (αn) ∈ λ } . We see that λ ⊂ λ×× =: (λ×)×; we say that λ is perfect if the equality holds. The normal cover of a subset S of λ is the subset of λ formed by the sequences of the M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1764 form (εnαn)n where (αn)n ∈ S and (εn)n ⊂ C with |εn| ≤ 1, for all n. We see that S is contained in its normal cover. S is said to be normal or solid if it coincides with its normal cover. For the general theory of locally convex spaces and Köthe sequence spaces, we refer the reader to [5]. Throughout this paper, λ will be a perfect (and then a normal) sequence space endowed with a normal bornology, that is a convex bornology having a basis S of solid sets, and for which the standard coordinate projections from λ to C are bounded. Following the terminology of [3], a sequence (xn)n ⊂ E is said to be totally λ−summable in E if there exists an absolutely convex element B ∈ B such that (xn)n ⊂ EB and (∥xn∥B)n ∈ λ. In other words, (xn)n = (αnbn)n, with (αn)n ∈ λ and {bn}∞n=1 ⊂ B. Starting from this definition, we introduce the vector valued sequence space λ(E) = { (xn)n ⊂ E : ∃B ∈ B, (xn)n ⊂ EB and (∥xn∥)n ∈ λ } . Due to the properties of B, the triangle inequality of the norms ∥ · ∥B and the fact that λ is normal, we see that λ(E) is a linear space. For S ∈ S and B ∈ B, we define S(B) = { (xn)n ⊂ EB, (∥xn∥B)n ∈ S } . 2. Properties of λ(E) In the sequel, the b-spaces E equipped with the convex bornology with basis B and λ with the normal bornology with basis S, will be supposed to be Hausdorff spaces. Starting from this setting, one can define, in a natural way, a convex bornology on λ(E) with basis S(B) by setting S(B) = { H ⊂ λ(E) : ∃S ∈ S, B ∈ B such that H = S(B) } . In view of the hypothesis made on S and B, S(B) is indeed a basis for a convex bornology on λ(E) for which λ(E) is a Hausdorff space. Lemma 1. For a fixed k ∈ N, denote by πk the projection from λ(E) on E defined by πk(x) = xk, for all x = (xn) ∈ λ(E). Then, πk is a bounded linear map. Proof. Let B ∈ B and S ∈ S and fix k ∈ N. Since the bornology of λ is normal, the set {αk : (αn)n ∈ S} is bounded in C, and then so is {∥xk∥ : (xn)n ∈ S(B)}. This means that {xk : (xn)n ∈ S(B)} is bounded in EB. Thus, πk is bounded. ■ M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1765 Proposition 1. The spaces λ and E can be identified with closed subspaces of λ(E). Proof. Let I : E −→ λ(E), t −→ te1, where t is at the first component. It is clear that I is linear and one to one. Let B ∈ B, and S ∈ S such that e1 ∈ S, then I(B) ⊂ S(B) and I is bounded. Inversely, I−1 : I(E) = Ee1 → E is the restriction of π1 to the subspace I(E), and then it is bounded by Lemma 1. It remains to show that I(E) is closed in λ(E). We have I(E) = ⋂ k ̸=1 π −1 k ({0}). Since E is supposed to be a Hausdorff space, then {0} is closed and so is I(E). Now, fix 0 ̸= x0 ∈ E and let g : λ −→ λ(E), α = (αn)n −→ (αnx0)n = αx0. It is clear that g is linear and one to one. Let S ∈ S, and B ∈ B with x0 ∈ B. Then, g(S) ⊂ S(B); so g is bounded. Inversely, if S ∈ S and B ∈ B, then g−1(S(B) ∩ λx0) = 1 ∥x0∥B S, and then g−1 : g(E) = λx0 → λ is bounded. It remains to show that g(λ) is closed in λ(E). Let {α(k)x0 = (α (k) n x0)n}∞k=1 be a sequence in λx0 which converges to x = (xn)n ∈ λ(E). By Lemma 1, {α(k) n x0}∞k=1 converges to xn in E, for every n. As, the subspace Cx0 of E is closed in E, xn must belong to Cx0. Then, there is α = (αn) such that x = (xn)n = αx0. It is easy to see that α ∈ λ. We conclude that λx0 is closed in λ(E). ■ Proposition 2. λ(E) is complete if and only if λ and E are complete. Proof. If λ(E) is complete, then so are λ and E by Proposition 1. Inversely, suppose that λ and E are complete. We only show that if B and S are Banach disks in E and λ respectively, then S(B) is a Banach disk in λ(E). To simplify the notations, we set F = λ(E), H = S(B) and π the gauge of H. Let {(xi)i}∞i=1 be a Cauchy sequence in (FH , π). We have∣∣∣∣∥∥(∥xin∥B)n∥∥S − ∥∥(∥xjn∥B)n∥∥S∣∣∣∣ ≤ ∣∣∣∣∥∥(∥xin∥B)n − (∥xjn∥B)n ∥∥ S ∣∣∣∣ ≤ ∥∥(∥xin∥B − ∥xjn∥B)n ∥∥ S ≤ ∥∥(∥xin − xjn∥B)n ∥∥ S = π((xi − xj)n). This means that {(∥xi∥B)i}∞i=1 is a Cauchy sequence in the complete space (λS , ∥ · ∥S); let α = (αn)n be its limit in λS . Fix n ∈ N. Due to the boundedness of the projections, {∥xin∥B}∞i=1 converges to αn and {xin}∞i=1 is a Cauchy sequence in the complete space EB; denote by xn its limit. Thus, ∥xn∥B = αn, and x = (xn)n ∈ λ(E). It remains to prove the convergence of {(xi)i}∞i=1 to x. This derives from the fact that {(∥xi − x∥B)i}∞i=1 is a Cauchy sequence in (λS , ∥ · ∥S) and its limit is nothing but the zero sequence in λ. ■ 3. Nuclearity of λ(E) A linear mapping f : E → F between complete normed spaces is said to be nuclear if there exist (εn)n ∈ ℓ1, a bounded sequence (an)n in the continuous dual E′ of E and a M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1766 bounded sequence (yn)n ⊂ F such that f(x) = ∞∑ n=1 εnan(x)yn, for all x ∈ E. A b-space E is said to be nuclear (a Schwartz space) if for every Banach disk A in E there is a Banach disk B ⊃ A in E such that the inclusion mapping EA → EB is nuclear (compact). Proposition 3. The tensor product λ⊗ E is identifiable with a subspace of λ(E). Proof. We see that for all α = (αn)n ∈ λ and x ∈ E, (αnx)n ∈ λ(E). Define the bilinear mapping φ : λ × E → λ(E), such that φ(α, x) = (αnx)n. There exists a linear mapping ℓ : λ ⊗ E → λ(E), with ℓ(α ⊗ x) = (αnx)n. Let us show that ℓ is one to one. Suppose that z ∈ λ⊗ E such that ℓ(z) = 0. We can write z = ∑k i=1(α i n)n ⊗ xi, for which {(αi n)n}ki=1 and {xi}ki=1 are linearly independent. But, ℓ(z) = k∑ i=1 ℓ(αi ⊗ xi) = k∑ i=1 (αi nxi)n = ( k∑ i=1 αi nxi ) n . Since ℓ(z) = 0 then (∑k i=1 α i nxi ) n = 0 and ∑k i=1 α i nxi = 0, for every n. But, as {xi}ki=1 is linearly independent, αi n = 0, for all 1 ≤ i ≤ k and n ∈ N. Thus, z = ∑k i=1(α i n)n ⊗ xi = 0, and ℓ is one to one. ■ Lemma 2. Let S and B be Banach disks in λ and E respectively, N(x) = ∥∥(∥xn∥B)n∥∥S for all x = (xn)n ∈ λS(EB) and N1(z) = N(ℓ(z)) for all z ∈ λS ⊗ EB. Then, 1. N1 is a cross-norm on λS ⊗ EB, that is N(α⊗ x) = ∥α∥S∥x∥B, for every α ∈ λS and x ∈ EB. 2. The mapping ℓ : λS ⊗ EB → λS(EB) is isometric and can be extended to a unique linear mapping ℓ̂ : λS⊗̂N1EB → λS(EB), where λS⊗̂N1EB the completion of the normed space (λS ⊗N1 EB, N1). Proof. Since N is a solid norm and ℓ is a one to one linear mapping, N1 is a norm. It is clear that N1(α⊗ x) = ∥α∥S∥x∥B, and 1. holds. By the definition of N1, we see that ℓ is isometric from λS ⊗EB to the complete space λS(EB), and then it has an extension to the completion λS⊗̂N1EB of λS ⊗N1 EB. This gives the second item. ■ We will make use of the following result to represent λ(E) as a bornological tensor product. Proposition 4. [4, Ch VIII, Prop. 4] 1. There is a convex bornology b on λ⊗ E (the finest one) making bounded the inclusion mappings λS ⊗N1 EB → λ(E). Moreover, λ⊗b E = lim−→λS ⊗N1 EB. 2. b is located between the projective bornology π and the injective bornology ε. 3. If λ or E is nuclear, then π = b = ε. 4. If λ and E are nuclear, the bornological completion λ⊗̃bE of λ ⊗b E is the inductive limit of the Banach spaces λS⊗̂N1EB. M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1767 Now, we prove Theorem 1. If λ and E are nuclear, the equality λ(E) = λ⊗̃bE holds algebraically and bornologically. Proof. Consider the linear mapping ℓ : λ⊗b E → λ(E) defined in the proof of Propo- sition 3. According to the definition of the norms N and N1, we see that ℓ is bounded, and since λ(E) is complete, ℓ can be extended to a bounded linear mapping ℓ̃ from the bornological completion λ⊗̃bE of λ⊗b E to λ(E). We will prove that ℓ̃ makes λ⊗̃bE and λ(E) bornologically isomorphic. Let z ∈ λ⊗̃bE be such that ℓ̃(z) = 0. By [4, Ch VIII, Prop. 2], a sequence {zk}∞k=1 of elements of λ⊗b E converges to z. Then {zk − z}∞k=1 is a null sequence in some subspace λS⊗̃bEB. Thus, ℓ̂(z) = ℓ̂(lim k ι(zk)) = lim k (ℓ̂ ◦ ι)(zk) = lim k ℓ(zk) = lim k (ℓ̃ ◦ ι)(zk) = ℓ̃(lim k zk) = ℓ̃(z) = 0. Here ι is the canonical injection from λ⊗b E to its completion λ⊗̃bE. By Lemma 2, ℓ̂ is isometric and then it is one to one, then z = 0, and ℓ̃ is one to one. We will prove that ℓ̃ is onto as follows. Let A ∈ B be a Banach disk; since E is nuclear we can select a Banach disk B ∈ B containing A such that the inclusion EA → EB is nuclear. There are (εk)k ∈ ℓ1, a bounded sequence (ak)k in the continuous dual (EA) ′ of EA and a bounded sequence (yk)k ⊂ EB such that x = ∞∑ k=1 εkak(x)yk, for all x ∈ EA. (1) Let x = (xn)n ∈ λS(EA), and αk = (αk n)n =: (ak(xn))n. We have |αk n| = |ak(xn)| ≤ ∥ak∥∥xn∥A ≤ ( sup p ∥ap∥ ) ∥xn∥A, for all k, n. (2) The sequence (ak)k being bounded in (EA) ′, supp ∥ap∥ is finite, αk = (αk n)n ∈ λS(EA), for all k, and, by (2), ∥αk∥S ≤ (supp ∥ap∥)∥(∥xn∥A)n∥S and then supk ∥αk∥S is finite. Then, r∑ k=1 N1(εkα k ⊗ yk) = r∑ k=1 |εk|∥αk∥S∥yk∥B ≤ (sup p ∥ap∥)(sup p ∥yp∥)N(x) r∑ k=1 εk. (3) As, λS(EB) is a complete normed spaces, the series ∑∞ k=1 εkα k ⊗ yk converges in λS(EB) to a limit g(x). Moreover, ℓ̃(g(x)) = x. (4) Indeed, if z = (zn)n ∈ λS(EB) is such that z = ℓ̃(g(x)), then z = (zn)n = ℓ̃ ( ∞∑ k=1 εk(ak(xn))n ⊗ yk ) = ∞∑ k=1 εk ℓ̃((ak(xn))n ⊗ yk) M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1768 = ∞∑ k=1 εkℓ((ak(xn))n ⊗ yk) = ∞∑ k=1 εk(ak(xn)yk)n. But the projections are bounded by Lemma 1, then zn = ∞∑ k=1 εkak(xn)yk, for all n. By (1), zn = xn, for all n, and ℓ̃(g(x)) = x. This means that ℓ̃ is onto. In the other hand, if K is bounded in λ(E), then K is contained and bounded in some λS(EB), and ℓ̃(g(K)) = K, from what, we conclude that the inverse of ℓ̃ is bounded. ■ We are now ready to prove the main result of this section. Theorem 2. Let E be a complete b-space and λ be a normal sequence space. Then λ(E) is nuclear if and only if λ and E are nuclear. Proof. If λ(E) is nuclear then, by Proposition 1, E and λ are closed subspaces of λ(E) and then they are nuclear also. Inversely, suppose that E and λ are nuclear. By Proposition 4, λ⊗̃bE is nuclear. So by Theorem 1, λ(E) is nuclear. ■ Theorem 3. Let E be a complete b-space and λ be a normal sequence space. (i) If λ is nuclear then, λ(E) is a Schwartz space if and only if E is a Schwartz space. (ii) If E is nuclear then, λ(E) is a Schwartz space if and only if λ is a Schwartz space. Proof. Suppose that E is nuclear. If λ(E) is a Schwartz space, then λ, being a closed subspace of λ(E) by Proposition 1, is a Schwartz space. Inversely, suppose that E is nuclear and λ is a Schwartz space. Let A ∈ B and S ∈ S be a Banach disks in E and λ respectively. Since E is nuclear we can select a Banach disk B ∈ B containing A such that the inclusion EA → EB is nuclear. So, there are (εk)k ∈ ℓ1, a bounded sequence (ak)k in the continuous dual (EA) ′ of EA and a bounded sequence (yk)k ⊂ EB such that x = ∞∑ k=1 εkak(x)yk, for all x ∈ EA. (5) Since λ is a Schwartz space, there is a Banach disk T in λ such that the injection λS → λT is compact. We will show that the injection λS(EA) → λT (EB) is compact. Let {xi = (xin)n}∞i=1 (6) M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1769 be a sequence in S(A). By (5), we have xin = ∞∑ k=1 εkak(x i n)yk, for all n, i. (7) The sequence (ak)k being bounded in (EA) ′, there is a constant c > 0 such that |ak(xin)| ≤ c∥xin∥A for all i, k, n. This means that {(ak(xin))n}∞i=1 ⊂ λS and that {(ak(xin))n}∞i=1 ⊂ cS. (8) A subsequence {(ak(xjn))n}∞j=1 of {(ak(xin))n}∞i=1 should converge in λT to αk = (αk n)n. In the other hand, the equation (8) shows that the sequence {(ak(xjn))n}∞k,j=1 is bounded in λS . For every n ∈ N, there cn > 0 such that for all j, k |ak(xjn)| ≤ cn and then |αk n| ≤ cn. (9) For every n ∈ N, since {αk nyk}∞k=1 is bounded in the complete normed space EB, the series ∑ k εkα k nyk converges to a limit xn ∈ EB. Let x = (xn)n. Since {(αk n)n}∞k=1 is bounded in λS and {yk}∞k=1 is bounded in EB, the sequence {(αk nyk)n}∞k=1 is bounded in λS(EB) and then in λT (EB). Thus, the series ∑ k εk(α k nyk)n converges in λT (EB) to z = (zn)n. Since the projections are bounded by Lemma 1, one has zn = ∑ k εkα k nyk for all n, and then x = z ∈ λT (EB). It remains to prove that {xj}∞i=1 converges in (λT (EB), N) to x. We have, xj − x = ∑ k εk(an(x j n)− αj n)nyk and N(xj − x) ≤ ∑ k |εk|∥(an(xjn)− αj n)n∥S∥yk∥B (10) For j, k, let βj k = ∥ak(xjn)− αk n∥T and γk = ∥yk∥B. (11) Then, (γk)k ∈ c0 and {(εkβj k)k} ∞ j=1 is a sequence in ℓ1 which is σ(ℓ1, c0)−bounded, then it has a convergent subsequence say, {(εkβr k)k}∞r=1. (12) As, lim r→∞ εkβ r k = 0, for all k, then the sequence in (12) converges to 0 in (ℓ1, σ(ℓ1, c0)). By (11) and (10), we have N(xr − x) ≤ ∑ k |εkβr k|γk, for all r ∈ N. Thus, {xr − x}∞r=1 converges to 0 in λT (EB), and (6) has a convergent subsequence. This finishes the proof of (i). The proof of (ii) is similar by interchanging the roles of E and λ in the proof. ■ M. A. Sidaty / Eur. J. Pure Appl. Math, 16 (3) (2023), 1762-1771 1770 4. Nuclearity of λ{E} Notice that a locally convex space is said to be nuclear (resp. a Schwartz space) if the convex bornology of equicontinuous subsets of its topological dual is nuclear (resp. of Schwartz). Let λ be a perfect sequence space and E a locally convex space whose topology is defined by a family M of absolutely convex equicontinuous subsets of its topological dual E′. Define λ{E} = {(xn)n ⊂ E : (PM (xn))n ∈ λ}, where PM (xn) = sup a∈M |a(xn)|. If a topology on λ is defined by family S of normal, absolutely convex and σ(λ×, λ)−bounded subsets of λ×, then a locally convex topology can be defined on λ{E} by the family of semi-norms (πS,M )S∈S,M∈M, such that, if x = (xn)n ∈ λ{E} then πS,M ((xn)n) = PS((PM (xn))) = sup{ ∞∑ n=1 |αnPM (xn)| : (αn)n ∈ S}. For the topology so defined, Ronald C. Rosier in [10] proved that the dual space (λ{E})∗ of λ{E} is λ×(E′) and that a subset of (λ{E})∗ is equicontinuous if and only if it is contained in some S(M) for S ∈ S and M ∈ M. Starting from this setting, Theorem 2 gives Theorem 4. λ{E} is nuclear if and only if λ and E are nuclear. Also, Theorem 3 gives Theorem 5. If E (resp. λ) is nuclear, then λ{E} is a Schwartz space if and only if λ (resp. E) is a Schwartz space. 5. Conclusion In this paper we have characterized the bornological structure, the completeness and the nuclearity of λ(E) in terms of that of λ and E. An application to the nuclearity of the locally convex space λ{E} is given. Acknowledgements The author is grateful to reviewers for their suggestions and comments which improved the quality of the paper. REFERENCES 1771 References [1] M. Florencio and Pedro J. Paúl. Una representación de cietros ϵ-productos tensori- ales. In Actas de las Jornadas Matematicas Hispano Lusas, Murcia., pages 191–203, Murcia, Spain, 1985. Universidad de Murcia. [2] M. Florencio and Pedro J. Paúl. La propiedad ak en ciertos espacios de suecsiones vec- toriales. In Dep. Mat. Univ. Extremadura, editor, Proc. Eleventh Spanish-Portuguese Conference on Mathematics, pages 197–203, 1986. [3] M. Florencio and Pedro J. Paúl. Barrelledness conditions on vector valued sequence spaces. Arch. Math., 48:153–164, 1987. [4] H. Henri-Hogbé-N’Lend. Théorie de Bornologies et Applications. Springer-Verlag, Lecture Notes, 213, Berlin and Heidelberg, 1971. [5] G. Köthe. Topological Vector Spaces I and II. Springer-Verlag, Berlin, Heidelberg, New York, 1979. [6] L. Oubbi and M. A. Ould Sidaty. Dual space of certain locally convex sequence spaces. Revista de la Real Academia de Ciencias de Zargoza, 59:79–88, 2004. [7] L. Oubbi and M. A. Ould Sidaty. Reflexivity of spaces of weakly summable sequences. Rev. R. Acad. Cien. Serie A. Mat., 101(1):51–62, 2007. [8] E. Pietsch. Verallgemeinerte Vollkommene Folgenräume. Akademie-Verlag, Berlin, Heidelberg, New York, 1962. [9] E. Pietsch. Nuclear locally convex spaces. Springer-Verlag, Berlin, Heidelberg, New York, 1972. [10] R. C. Rosier. Dual space of certain vector sequence spaces. Pacific J. Math., 46(2):487–501, 1973. [11] M. A. Ould Sidaty. Reflexivity and AK-property of certain vector sequence spaces. Bull. Belg. Math. Soc., 10(4):579–583, 2003. [12] M. A. Ould Sidaty. Nuclearity of certain vector-valued sequence spaces. Rev. Real Academia de Ciencias. Zaragoza., 62:81–89, 2007. [13] M. A. Ould Sidaty. Reflexivity of vector-valued Köthe-Orlicz sequence spaces. Turk J Math., 42(3):911–923, 2018.