= x− x(p) = (0, 0, . . . , 0, xp+1, xp+2, . . . ). If en is the nth unit coordinate vector of CN, then x(p) = ∑p n=1 xnen. We will denote by λ(E)r (resp. λ⟨E⟩r), the subspace of λ(E) (rep. λ⟨E⟩) consisting of all the sequences x = (xn)n which are limits of their finite sections x(p). The reader is referred to [6, 14] for notations and concepts related to the Köthe theory of sequence spaces and the general theory of locally convex spaces. 2. Bounded sets of λ(E) If B is a closed, absolutely convex and bounded subset of E, and S = Bλ∗ , let B̃ = { (xn)n ∈ λ(E) : ∀α = (αn)n ∈ S, ∑ n αnxn ∈ B } . (1) We have the following result. Proposition 1. The collection {B̃ : B bounded in E} constitutes a fundamental system of bounded sets for λ(E). Proof. If B is a bounded set in E, then the corresponding set B̃ in (1) is bounded in λ(E), by [8, Proposition 1]. Now, let B be a bounded set of λ(E) and S the unit ball of λ∗. Consider the subset B of E defined by B = { y ∈ λ(E) : y = ∞∑ n=1 αnxn, for some α ∈ S and x = (xn)n ∈ B } . M. A. Sidaty / Eur. J. Pure Appl. Math, 17 (1) (2024), 171-179 174 Let a ∈ E′, M ∈ M with a ∈ M and x = (xn)n ∈ B. Then,∣∣∣∣∣ 〈 a, ∞∑ n=1 αnxn 〉∣∣∣∣∣ = ∣∣∣∣∣ ∞∑ n=1 αna(xn) ∣∣∣∣∣ ≤ ∞∑ n=1 |αna(xn)| ≤ εM (x). Since S is a normal disk in λ∗ and B is bounded in λ(E), then B is a bounded disk in E. Moreover, by the definition of B, we see that B ⊂ B̃. ■ Lemma 2. For every t ∈ E and β = (βn)n ∈ λ, we have (βnt)n ∈ λ⟨E⟩. Proof. For t ∈ E, let δt denote the evaluation defined on E′ by δt(x ′) = x′(t). We have, if M ∈ M, then |δt(x′)| ≤ PM (t)∥x′∥M for every x′ ∈ E′ M . This means that δt ∈ (E′ M )′ and that ∥δt∥ ≤ PM (t). Let β = (βn)n ∈ λ and (an)n ∈ λ∗[E′ M ]. By the definition of λ∗[E′ M ], (an(t))n = (δt(an)) ∈ λ∗, and then ∞∑ n=1 |an(βnt)| = ∞∑ n=1 |an(t)βn| < ∞. Thus, (βnt)n ∈ λ⟨E⟩. ■ Now, for S = Bλ and a closed absolutely convex bounded subset B of E, define B̄ = { ∞∑ k=1 ξkβ kxk : βk = (βk n)n ∈ Bλ, xk ∈ B, and ∞∑ k=1 |ξk| ≤ 1 } . (2) Proposition 3. The set B̄ is a bounded subset of λ⟨E⟩. Proof. Let {βk = (βk n)n}∞k and {xk}∞k be sequences in Bλ and B respectively. Fix k ∈ N, M ∈ M and a = (an)n ∈ λ∗[E′ M ] = λ∗(E′ M ), with ∥a∥λ∗(E′ M ) ≤ 1. As in the proof of the previous lemma, δxk denotes the evaluation defined on E′ M . We have ∞∑ n=1 ∣∣∣an(βk nxk) ∣∣∣ = ∞∑ n=1 ∣∣∣βk nan(xk) ∣∣∣ = ∞∑ n=1 ∣∣∣βk nδxk (an) ∣∣∣ = ∥βk∥λPM (xk) ∞∑ n=1 ∣∣∣∣ βk n ∥βk∥λ δxk PM (xk) (an) ∣∣∣∣ ≤ ∥βk∥λPM (xk)∥a∥λ∗(E′ M ) ≤ ∥βk∥λPM (xk) ≤ PM (xk). Since B is bounded in E, then there exists ρ > 0, so that PM (xk) ≤ ρ for every k ∈ N; and, by the definition of σM , one has σM (βkxk) ≤ ρ, for every k ∈ N. Moreover, if (ξk)k satisfies ∑∞ k=1 |ξk| ≤ 1 then, ∞∑ k=1 σM (ξkβ kxk) ≤ ∞∑ k=1 |ξk|∥βk∥λPM (xk) ≤ ρ ∞∑ k=1 |ξk| ≤ ρ. (3) By Lemma 2, the terms of the series ∑∞ k=1 ξkβ kxk belong to λ⟨E⟩. Since λ⟨E⟩ is sequen- tially complete, we derive from (3) that this series is convergent in λ⟨E⟩ and that the corresponding set B̄ in (2) is well defined, contained and bounded in λ⟨E⟩. ■ M. A. Sidaty / Eur. J. Pure Appl. Math, 17 (1) (2024), 171-179 175 3. Köthe Duals of λ(E) and λ⟨E⟩ Following [4], if F is a linear subspace of EN, the generalized Köthe dual of F is defined by F ∗ = { (an)n ⊂ E′ : ∑ |an(xn)| converges for all x = (xn)n ∈ F } . For every x ∈ E, denote by δx the evaluation defined, as in the proof of Lemma 2, by δx(x ′) = x′(x), for x′ ∈ E′. Thanks to the linear and isometric map δ : x → δx from E to E′′, we always have F ⊂ F ∗∗. The sequence space F is said to be perfect if F ∗∗ = F . Proposition 4. Let λ be a perfect normed sequence space with dual space λ∗ and E a locally convex space. Then (λ(E)r) ∗ = (λ(E))∗ and (λ⟨E⟩r)∗ = (λ⟨E⟩)∗. Proof. We prove that (λ(E)r) ∗ = (λ(E))∗, the same argument applies for the second equality. It is clear that (λ(E))∗ ⊂ (λ(E)r) ∗. Let a = (an)n ∈ (λ(E)r) ∗ and x = (xn)n ∈ λ(E). To prove that ∑∞ n=1 |an(xn)| converges, it is enough to prove that, for every (γn)n ∈ c0, the series ∑∞ n=1 |γnan(xn)| converges. Set y = (yn)n where yn = γnxn, for all n ∈ N. We see that y ∈ λ(E). In the other hand, for M ∈ M, a ∈ M , α = (αn)n ∈ Bλ∗ and p ∈ N, one has ∞∑ n=p+1 |αna (γnxn)| ≤ sup n≥p+1 |γn| ∞∑ n=p+1 |αna (xn)| ≤ ∥γ
∥c0ϵM (x). This shows that ϵM (y
) ≤ ∥γ
∥c0ϵM (x), and then y ∈ λ(E)r since (γ
)p converges to 0. Now, we have ∞∑ n=1 |γnan(xn)| = ∞∑ n=1 |an(γnxn)| = ∞∑ n=1 |an(yn)| < ∞. This completes the proof. ■ According to [7, Theorem 7], the continuous dual λ(E)r of (λ(E)r) ′ is given by (λ(E)r) ′ =⋃ M∈M λ∗⟨E′ M ⟩. In particular, if λ and E are Banach spaces then (λ(E)r) ′ = λ∗⟨E′⟩. (4) The last equality is actually topological, by ([6, 15.12(2)]). Proposition 5. For every Banach space E, we have (a) the Köthe dual of (λ(E))∗ satisfies (λ(E))∗ = λ∗⟨E′⟩ = (λ(E)r) ′, (b) the Köthe dual of (λ(E))∗ satisfies (λ(E))∗∗ = λ(E′′). In particular, if E is reflexive then (λ(E))∗∗ = λ(E). M. A. Sidaty / Eur. J. Pure Appl. Math, 17 (1) (2024), 171-179 176 Proof. By the definition of the spaces λ∗⟨E′⟩ and (λ(E))∗, we have λ∗⟨E′⟩ ⊂ (λ(E))∗. Let a = (an)n ∈ (λ(E))∗. Using the closed graph theorem ([6, 15.12(3)]), we can prove that the mapping fa : λ(E)r → ℓ1 defined by fa(x) = (an(xn))n is continuous, and then a ∈ (λ(E)r) ′. So, (λ(E))∗ ⊂ (λ(E)r) ′. The part (a) follows from (4). For (b), we have (λ(E))∗ = (λ(E)r) ∗, (by Proposition 4) = (λ(E)r) ′, (by (a)) = (λ⊗̃εE)′, (by [3, Prop. 2]) = λ∗⊗̃πE ′, (by [6, 45.6(5)]). On the other hand, since (λ∗⟨E′⟩)′ = (λ⊗̃εE)′′ = (λ∗⊗̃πE ′)′ = L(λ∗, E′′), (by [6, 41.3(6)]) = λ(E′′), (by [10, Propoition 2]) then, (λ(E))∗∗ = λ(E′′). ■ Now, by [8, Theorem 1], the continuous dual (λ⟨E⟩r)′ of λ⟨E⟩r is given by the algebraic equality (λ⟨E⟩r)′ = ⋃ M∈M λ∗(E′ M ). If λ and E are Banach spaces then (λ⟨E⟩r)′ = λ∗(E′). (5) This equality is topological, by ([6, 15.12(2)]). Similarly, we have Proposition 6. For every Banach space E, the following equalities hold (a) (λ⟨E⟩)∗ = λ∗(E′) = (λ⟨E⟩r)′, (b) (λ⟨E⟩)∗∗ = λ⟨E′′⟩. In particular, if E is reflexive then (λ⟨E⟩)∗∗ = λ⟨E⟩. Proof. The proof is similar to that of Proposition 5, but we present it for the sake of completeness. By [5, Theorem 1], λ∗(E′) = λ∗[E′], and then, by the definition of the space λ⟨E⟩, we have λ∗(E′) ⊂ (λ⟨E⟩)∗. On the other hand, in view of the closed graph theorem ([6, 15.12(3)]), we deduce that every a = (an)n ∈ (λ⟨E⟩)∗ corresponds to a continuous linear form on λ⟨E⟩r by setting fa(x) = ∑∞ n=1 an(xn). Thus, (λ⟨E⟩)∗ ⊂ (λ⟨E⟩r)′. The part (a) follows from (5). Regarding (b), we have (λ⟨E⟩)∗ = (λ⟨E⟩r)∗, (by Proposition 4) = (λ⟨E⟩r)′, (by (a)) = λ∗(E′), (by [8, Theorem 1]). This leads to (λ⟨E⟩)∗∗ = (λ∗(E′))∗ = λ∗∗⟨E′′⟩ = λ⟨E′′⟩, (by (a) of Proposition 5). This ends the proof. ■ M. A. Sidaty / Eur. J. Pure Appl. Math, 17 (1) (2024), 171-179 177 Proposition 7. Suppose that E and λ are Banach spaces with λ reflexive. Then, λ∗⟨E′⟩r = λ∗⟨E′⟩. Proof. Let a = (an)n ∈ λ∗⟨E′⟩, and consider φa : λ(E′′) → ℓ1, φa((x ′′ n)n) = (x′′n(an))n. The linear mapping φa is well defined, since λ(E′′) = λ[E′′] by [5, Theorem 1]. Let us show that φa is weak to weak continuous. If (αn)n ∈ ℓ∞ then (αnan)n ∈ λ∗⟨E′⟩ and for all (x′′n)n ∈ λ(E′′), we have〈 (αn)n, (x′′n(an))n 〉 = 〈 (αnan)n, (x ′′ n)n 〉 . Let B be a bounded set in λ(E)r. The Alaoglu-Bourbaki Theorem ([6, 20.9(4)]) asserts that B is relatively weak∗ compact. We derive from [6, 22.4(3)] that { (an(xn))n : (xn)n ∈ B } is relatively compact in ℓ1 and then lim p→∞ sup ∞∑ n=p+1 |an(xn)| : (xn)n ∈ B = 0. This means that (a
)p is a null sequence in λ∗⟨E′⟩. This completes the proof. ■ Proposition 8. Let E be a Banach and λ a reflexive Banach sequence space. Then, the elements of λ∗⟨E′⟩ are the sequences a = (an)n ⊂ E′ that have the form a = ∞∑ k=1 λkβ kx′k, where (λk)k ∈ ℓ1 and, for every k ∈ N, βk = (βk n)n ∈ Bλ and x′k ∈ BE′. Proof. As in Proposition 7, we have λ∗⟨E′⟩ = (λ(E)r) ′, by [7, Theorem 7] = (λ⊗̃εE)′, by [3, Prop. 2] = I(λ× E), integral bilinear forms on λ× E, by [6, 45.1(2)] = LI(λ,E′), integral mappings of λ in E′, by [6, 45.4(1)] = N (λ,E′). nuclear operators from λ in E′ by [6, 45.6(1) and 45.6(4)] For a = (an)n ∈ λ∗⟨E′⟩, the corresponding fa ∈ LI(λ,E′) is defined by fa(α) ∈ E′ such that fa(α)(t) = B(α, t) = ⟨a, αt⟩, for α = (αn)n ∈ λ and t ∈ E, where B ∈ I(λ × E) is the integral bilinear form on λ× E corresponding to fa. Now, since fa ∈ N (λ,E′), then by [6, 42.5(5)-(6)], there are (λk)k ∈ ℓ1, a sequence M. A. Sidaty / Eur. J. Pure Appl. Math, 17 (1) (2024), 171-179 178 {βk = (βk n)n : k ∈ N} in Bλ and a sequence (x′k)k in BE′ such that, for every α ∈ λ and t ∈ E, we have ⟨a, αt⟩ = fa(α)(t) = ∞∑ k=1 λk⟨βk, α⟩x′k(t). This implies that an = ∑∞ k=1 λkβ k nx ′ k for every n ∈ N and that a = ∑∞ k=1 λkβ kx′k. By ascending the previous chain of equalities, we easily see that the inverse is true. ■ Proposition 9. (λ(E))∗ = (λ(E))′ if and only if λ(E)r = λ(E). Proof. By (a) of Proposition 5, if λ(E)r = λ(E) then (λ(E))∗ = (λ(E)r) ′ = (λ(E))′. Inversely, suppose that (λ(E))∗ = (λ(E))′. As in the proof of Proposition 7, for every x = (xn)n ∈ λ(E), φx : (λ(E))′ → ℓ1, φx((an)n) = (an(x))n, is well defined, linear and weak to weak continuous. Denote by H the closed unit ball of (λ(E))∗. Here also, the Alaoglu-Bourbaki Theorem ([6, 20.9(4)]) guarantees that H is relatively weak∗ compact. We derive from [6, 22.4(3)], that { (an(xn))n : (an)n ∈ H } is relatively compact in ℓ1 and then lim p→∞ εM (x
) = lim p→∞ sup ∞∑ n=p+1 |an(xn)| : (an)n ∈ H = 0. Thus, x ∈ λ(E)r. ■ Proposition 10. (λ⟨E⟩)∗ = (λ⟨E⟩)′ if and only if λ⟨E⟩r = λ⟨E⟩. Proof. The proof is similar to that of Proposition 9 when interchanging the roles of λ(E) and (λ(E))′ by those of λ⟨E⟩ and (λ⟨E⟩)′, respectively. ■ Conclusion Let E be a Banach space and λ a perfect Banach sequence space which is reflexive. We prove that λ(E) and λ⟨E⟩ have the AK property if and only if the Köthe dual and the continuous dual are equal. If, moreover E is reflexive, these spaces become perfect. Acknowledgements I wish to thank the reviewers for their remarks and suggestions which improved the presentation of the paper. REFERENCES 179 References [1] H. Apiola. Duality between spaces of p-summing operators and characterization of nuclearity. Math. Ann., 219:53–64, 1974. [2] J. S. Cohen. Absolutely p-summing, p-nuclearoperators and their conjugates. Math. Ann., 201:177–200, 1973. [3] M. Florencio and Pedro J. Paúl. Una representación de cietros ϵ-productos tensoriales. In Actas de las Jornadas Matematicas Hispano Lusas, Murcia, pages 191–203, Murcia, Spain, 1985. Universidad de Murcia. [4] M. Florencio and Pedro J. Paúl. Barrelledness conditions on vector valued sequence spaces. Arch. Math., 48:153–164, 1987. [5] M. Florencio and Pedro J. Paúl. A note on λ-multiplier convergent series. Časopis P̌est. Mat., 113:421–428, 1988. [6] G. Köthe. Topological Vector Spaces I and II. Springer-Verlag, Berlin, Heidelberg, New York, 1979. [7] 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. [8] 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. [9] E. Pietsch. Nuclear locally convex spaces. Springer-Verlag, Berlin, Heidelberg, New York, 1972. [10] M. A. Ould Sidaty. Reflexivity and AK-property of certain vector sequence spaces. Bull. Belg. Math. Soc., 10(4):579–583, 2003. [11] M. A. Ould Sidaty. Nuclearity of certain vector-valued sequence spaces. Rev. Real Academia de Ciencias. Zaragoza., 62:81–89, 2007. [12] M. A. Ould Sidaty. Reflexivity of vector-valued Köthe-Orlicz sequence spaces. Turk. J. Math., 42(3):911–923, 2018. [13] M. A. Ould Sidaty. Nuclearity of a class of vector-valued sequence spaces. Eur. J. Pure Appl. Math., 16(3):1762–1771, 2023. [14] M. Valdivia. Topics on locally convex spaces. North-Holland, Amsterdam, New York, Oxford, 1982.