The dual of the sequence spaces with mixed norms l(p,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{(p, \infty )}$$\end{document}

We identify a norm-dense subspace of the dual of the sequence space l(p,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^{(p,\infty )}$$\end{document}, thus closing the existing gap in the literature. We based our approach on the notion of James orthogonality, absolutely continuous norms and on the uniform convexity and the uniform smoothness of the underlying subspaces.

The duality question for l P has been studied in [1] for p 1 , p 2 < ∞. More precisely, [1,Theorem 1.(a) in Section 3], one has l P * = l P when 1 ≤ p 1 , p 2 < ∞. In particular, it is easy to verify that l P is a Banach function space [8] of functions over N × N whose associate space l P is l P (see [2,7,8] for a detailed discussion). This paper will focus on the duality question for the case P = ( p 1 , ∞) when 1 ≤ p 1 < ∞. In this setting, the corresponding norm is: Our main result is Theorem 2.10, in which we show that a set of relatively simple linear functionals is dense in (l ( p 1 ,∞) ) * .
We recall the notion of absolutely continuous norm in our case (see [2,8] for the general setting): Definition 1.2 Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ and set P = ( p, q). An element f ∈ l P is said to have absolutely continuous norm if for any sequence of sets (E j ) in 2 N×N such that 1 E j → 0 pointwise as j → ∞, it holds that f 1 E j P → 0 as j → ∞.
The set of all elements in l P with absolutely continuous norm will be denoted with l P a . Note that l P a is a closed subspace of l P [2, Theorem I. 3.8].
The following statements will be of importance in the sequel.

Lemma 1.3
The following are equivalent and true for P = ( p, q) with 1 < p < ∞ and 1 < q < ∞: (i) l P a =l P , (ii) l P * = l P = l P .
Proof See [2,8] for a detailed proof of this Lemma.

A bounded linear functional
∈ X * is said to be norm-attaining if there exists x 0 ∈ B X such that We next recall the notions of uniform convexity and uniform smoothness in Banach spaces, these concepts will be at the heart of our approach. The interested reader will find a detailed survey on the geometry of Banach spaces, as well as further references, in [5]. To measure the degree of convexity of a Banach space X , its modulus of convexity δ X : [0, 2] → [0, 1] is defined by the quantity: Note that the same function is obtained if the infimum is taken over all x, y ∈ X with x = y = 1 and x − y = ε. We say that the Banach space X is uniformly convex if for all ε ∈ (0, 2], δ X (ε) > 0. The modulus of smoothness of X is the function ρ X : (0, ∞) → [0, ∞) defined by: The space X is called uniformly smooth if lim ε→0 ρ X (ε)/ε = 0.
We recall that X is uniformly smooth if and only if We underline the fact that X is uniformly convex (respectively, uniformly smooth) if and only if X * is uniformly smooth (respectively, uniformly convex), [5].
It is well known that the space l p is uniformly convex and uniformly smooth for p ∈ (1, ∞).

The dual space of ( p,∞)
It is easy to see that any linear functional ∈ (l ( p,∞) ) * can be written as where a ∈ l ( p,∞) (i.e, a can be identified with an element of the associate space of l ( p,∞) ) and d vanishes on the subspace of l ( p,∞) that consists of elements that have absolutely continuous norm, henceforth denoted by l ( p,∞) a . Since, for 1 < p < ∞, the associate space of l ( p,∞) is l We next introduce the following definitions. is referred to as the norm of discontinuity of f.
The following properties are obvious by definition: ) * stands for the subset of (l ( p,∞) ) * consisting of linear functionals L such that Next the notion of James orthogonality will be recalled.

Definition 2.4
Let (X, · ) be a Banach space, x, y ∈ X . An element 0 = x ∈ X is said to be j-orthogonal to y, denoted by x ⊥ X y (or x ⊥ j y) iff the inequality x ≤ x + αy holds for any α ∈ R.
Note that James' orthogonality on a Banach space is not symmetric. An observation is in order at this point: We also emphasize the fact that if the norm of X is uniformly smooth, then, for 0 = x ∈ X , the set is a vector subspace of X and codim(x ⊥ j ) = 1. In particular, for 1 < p < ∞ and a, b ∈ l p , it follows that for Indeed, (2.1) can be obtained by observing that the derivative of the real-valued function and that in view of the orthogonality condition, it should vanish at t = 0. Next we adapt the definition of James orthogonality to the case under consideration.
. For given f ∈ l ( p,∞) we define the James orthogonal complement of f as the vector subspace It is easily seen that codim(f ⊥ j ) = 1.
It is important to mention that the validity of codim(f ⊥ j ) = 1 results from each space in the sequence (i.e l p ) having uniformly smooth norm (1.2), and that the uniqueness of the subspace f ⊥ j is guaranteed by the uniform convexity (1.1) of the norm of each space in the given sequence.
Note that f ⊥ j g iff for each i ∈ N we have: For the sake of clarity, we recall that a set function is a finitely additive measure if −∞ < μ(N) < ∞, μ is finitely additive and It is well known that (l ∞ ) * can be naturally identified with the space ba(N) of all finitely additive measures of finite total variation, via the one-to-one assignment where 1 A stands for the indicator function of the set A [4]. Given a finitely additive measure μ ∈ ba(N) it is customary to denote the linear functional λ ∈ (l ∞ ) * ) associated to μ in the above sense, by N ·dμ and its action on a = (a i ) ∈ l ∞ with N {a i } i dμ.
Notice that for each i ∈ N, Let μ ∈ ba(N) and define L ∈ (l ( p,∞) ) * by It is clear that L can be regarded as the composition of two linear maps, namely Obviously L is an element of (l ( p,∞) ) * and We underline the fact that for h i ∈ l p we have (Definition 2.6 and (2.2)) (2.2)) and if h i = 0, the norm of the bounded linear functional on l p associated to h i (2.2) is reached at Observe that for a given f ∈ l ( p,∞) , any g ∈ l ( p,∞) can be decomposed as with f⊥ j h. This can be achieved by setting, for each i ∈ N, . We will use the notation For F ∈ (l ( p,∞) ) * one obviously has F(g) = F({α i f i } i ) + F(h); this can also be written as Furthermore, let f i l p = 1 or 0 for each i and assume On account of the uniform smoothness of l p and of the fact that for each i ∈ N, f i ⊥ j h i (see (2.1)), for each i it must hold that: where 0 ≤ c(t) 0 as t → 0 and c(t) depends only on the uniform smoothness of l p (see connection with (1.2)). It follows that In all, which by virtue of (2.6) leads to: Letting t → 0 + , c(t) 0 and it readily follows that L(h) ≤ 0. A similar argument (letting t → 0 + , c(t) 0) yields L(h) ≥ 0. In all, L(h) = 0 as claimed. Then: (i) It can be assumed without loss of generality that j : ∞) and due the maximality of f we get L(h) = L(f).
(ii) If f is as given and Since L vanishes on l ( p,∞) a , one easily concludes where g = (f N , f N +1 , ...). In this case, it would follow which is a contradiction.
Next, for arbitrary g ∈ l ( p,∞) the decomposition given in (2.4) holds, namely with f ⊥ j h. By virtue of Lemma 2.8 and the linearity of L, one readily obtains Theorem 2.10 Let 1 < p < ∞. The subset of (l p,∞ ) * consisting of functionals of the form (2.3) is norm-dense in (l p,∞ ) * .
Proof The theorem is an immediate consequence of Theorem 2.9 in concert with the theorem of Bishop-Phelps [3].

Open problems and further comments
We conclude the paper with a generalization of the previous section results and some open problems related to l { p n } . In the proof of Theorem 2.9 the essential requirement is that all l p spaces are uniformly smooth and uniformly convex Banach spaces. Replacing spaces l p by a sequence of uniformly convex spaces for which exist a suitable uniform upper bound for the quantity c(t) in inequality (2.6) one can easily generalize the above arguments and derive to the next theorem.
is norm dense in (l ∞ ({X i })) * . In (3.1), for each i, λ i ∈ X * i , μ ∈ ba(N) and the notation in the paragraph preceding Example 2.7 has been used.
In connection with our work, we highlight Theorem 3.2.4 in [6], which deals with a similar result for an order continuous Koethe function space E over a probability space and a Banach space X ." Though the space l ( p,∞) we consider in our work can be described in the language of Koethe function spaces as E(X ), where E = l ∞ and X = l p , Theorem 3.2.4 above does not apply in this case, since l ∞ is not order continuous, which is easily seen by considering the sequence x 1 = (1, 1, ...1, ...), x 2 = (1/2, 1/2, 1, 1....), ...x n = (1/n, 1/n, ... ..1/n, 1, 1...), .... The essential novelty of our paper is the consideration of duality in the absence of order continuity. The space l p,∞ whose dual we compute in our work is not covered by Theorem 3.2.4 above and our result does not follow from the work by Pei Kee Lin.
We conclude this section by considering an example of a sequence space in which modulus of smoothness can naturally "deteriorated" as n → ∞.
Let It is well known that furnished with this norm l { p n } is a Banach space, separable if the sequence { p n } is bounded and reflexive if inf n p n > 1 (see [8]) In the sequel we will set p n = p n p n −1 if p n > 1, p n = ∞ if p n = 1. ∞). Then the following are equivalent: On another note, it was realized by Orlicz in 1931 [9] that for an unbounded sequence { p n }, if p n tends to infinity fast enough, then l { p n } is isomorphic to l ∞ . More precisely:  The characterization of the dual in the latter case remains open: a result in this direction would shed light on the description of the dual space of L p(·) ( ) for a domain ⊂ R n and a measurable p : → (1, ∞) with sup p(x) = ∞, which is hitherto unknown.
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