Sequence Lorentz Spaces and Their Geometric Structure

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity, and strict convexity in the sequence Lorentz spaces γp,w.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{p,w}.$$\end{document} Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space γ1,w.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{1,w}.$$\end{document} We also establish a complete description up to isometry of the dual and predual spaces of the sequence Lorentz spaces γ1,w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{1,w}$$\end{document} written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of γ1,w\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{1,w}$$\end{document} to one-complemented subspaces of γ1,w.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{1,w}.$$\end{document}

authors have studied the biduals and order continuous ideals of the Marcinkiewicz spaces for the pure atomic measure. The next significant paper was published in 2009 [10], in which there has been investigated, among others, strict monotonicity, smooth points, and extreme points with application to one-complemented subspaces. For other results concerning the issue devoted to one-complemented subspaces please see, e.g., [7,8,11].
The purpose of this article is to explore geometric properties of the sequence Lorentz spaces γ p,w and its dual and predual spaces. It is worth mentioning that we present an application of geometric properties to a characterization of one-complemented subspaces in the Lorentz spaces γ p,w in case of the pure atomic measure. It is necessary to mention that a characterization of geometric structure of the sequence Lorentz and Marcinkiewicz spaces does not follow immediately as a consequence of well-known results from the case of nonatomic measure in general.
The paper is organized as follows. In Sect. 2, we present the needed terminology. In Sect. 3, we show an auxiliary result devoted to a relationship between the global convergence in measure of a sequence (x n ) ⊂ 0 and the pointwise convergence of its sequence of decreasing rearrangements (x * n ). In case of the pure atomic measure, we also establish a correspondence between an identity of signs of the values for two different sequences in 0 and an additivity of the decreasing rearrangement operation for these sequences. Section 4 is devoted to an investigation of geometric structure of sequence Lorentz spaces γ p,w . Namely, we focus on complete criteria for order continuity and the Fatou property in Lorentz spaces for the pure atomic measure. Next, we present a characterization of strict monotonicity and strict convexity of γ p,w written in terms of the weight sequence w. In spirit of the previous result, we describe an equivalent condition for extreme points of the unit ball in the sequence Lorentz space γ 1,w . In Sect. 5, we solve the essential problem showing a full description of the dual and predual spaces of the sequence Lorentz space γ 1,w . First, we answer a crucial question under which condition does an isometric isomorphism exist between the dual space of the sequence Lorentz space γ 1,w and the sequence Marcinkiewicz space m φ . Next, we discuss complete criteria which guarantee that the predual space of the sequence Lorentz space γ 1,w coincides with the sequence Marcinkiewicz space m 0 φ . Additionally, we investigate necessary condition for the isometry between the predual of γ 1,w and the Marcinkiewicz space m 0 φ . In Sect. 5, we present an application of geometric properties of the sequence Lorentz space γ 1,w to a characterization of one-complemented subspaces. Namely, using an isometry between the classical Lorentz space d 1,w and the Lorentz space γ 1,w , we prove that there exists norm one projection on any nontrivial existence subspace of γ 1,w . Additionally, by the previous investigation and in view of [10], we establish a full characterization of smooth points in the sequence Lorentz space γ 1,w and its predual and dual spaces. Finally, we study an equivalent condition for extreme points in the dual space of the sequence Lorentz space γ 1,w .

Preliminaries
Let R, R + , and N be the sets of reals, nonnegative reals, and positive integers, respectively. A mapping φ : N → R + is said to be quasiconcave if φ(t) is increasing and φ(t)/t is decreasing on N and also φ(n) > 0 for all n ∈ N. Denote by 0 the set of all real sequences, and by S X (resp. B X ) the unit sphere (resp. the closed unit ball) in a Banach space (X , · X ). Let us denote by (e i ) ∞ i=1 a standard basis in R ∞ . A quasi-Banach lattice E ⊂ 0 equipped with a quasi-norm · E is called a quasi-Banach sequence space (or a quasi-Köthe sequence space) if the following conditions hold (1) If x ∈ 0 , y ∈ E, and |x| ≤ |y|, then x ∈ E and x E ≤ y E .
(2) There exists a strictly positive x ∈ E. For simplicity let us use the short symbol E + = {x ∈ E : x ≥ 0}. An element x ∈ E is called a point of order continuity, shortly x ∈ E a , if for any sequence (x n ) ⊂ E + such that x n ≤ |x| and x n → 0 pointwise we have x n E → 0. A quasi-Banach sequence space E is said to be order continuous, shortly E ∈ (OC), if any element x ∈ E is a point of order continuity. Given a quasi-Banach sequence space E is said to have the Fatou property if for all (x n ) ⊂ E + , sup n∈N x n E < ∞ and x n ↑ x ∈ 0 , then x ∈ E and x n E ↑ x E (see [2,16]). We say that E is strictly monotone if for any x, y ∈ E + such that x ≤ y and x = y we have The distribution for any sequence x ∈ 0 is defined by For any sequence x ∈ 0 its decreasing rearrangement is given by In this article we use the notation x * (∞) = lim n→∞ x * (n). For any sequence x ∈ 0 we denote the maximal sequence of x * by It is easy to notice that for any point x ∈ 0 , x * ≤ x * * , x * * is decreasing and subadditive. For more details of d x , x * , and x * * see [2,14]. We say that two sequences x, y ∈ 0 are equimeasurable, shortly x ∼ y, if d x = d y . A quasi-Banach sequence space (E, · E ) is called symmetric or rearrangement invariant (r.i. for short) if whenever x ∈ 0 and y ∈ E such that x ∼ y, then x ∈ E and x E = y E . The fundamental sequence φ E of a symmetric space E we define as follows φ E (n) = χ {i∈N:i≤n} E for any n ∈ N (see [2]). Let 0 < p < ∞ and w = (w(n)) n∈N be a nonnegative real sequence and let for any n ∈ N For short notation the sequence w is called a nonnegative weight sequence. In the whole paper, unless we say otherwise we suppose that w a nonnegative weight sequence is nontrivial, i.e., there is n ∈ N such that w(n) > 0. Let 0 < p < ∞. Now, we recall the sequence Lorentz space d p,w which is a subspace of 0 such that for any sequence x = (x(n)) n∈N ∈ d p,w we have It is well known that the Lorentz space d p,w is a norm space if and only if w is decreasing (see [17]). Furthermore, for any 0 < p < ∞, · γ p,w is a quasi-norm if and only if W satisfies condition Δ 2 , that is W (2n) ≤ K W (n) for any n ∈ N and for some K > 0 (see [13,20]). Additionally, if W satisfies condition Δ 2 and W (∞) = ∞, then for any 0 < p < ∞ the space d p,w is a separable r.i. quasi-Banach sequence space (see [13]). Recall, the sequence Lorentz space γ p,w is a collection of all real sequences x = (x(n)) n∈N such that Let us notice that for any nonnegative sequence w = (w(n)) n∈N the sequence Lorentz space γ p,w is a r.i. (quasi-)Banach sequence space equipped with the (quasi-)norm · γ p,w . Additionally, note that the space γ p,w is a Banach space if 1 ≤ p < ∞. It is easy to observe that the fundamental sequence of the Lorentz space γ p,w is given by φ γ p,w (n) = χ {i≤n,i∈N} γ p,w = (W (n) + W p (n)) 1/ p for every n ∈ N. Clearly, since x * ≤ x * * , we have γ p,w ⊂ d p,w for any 0 < p < ∞. Moreover, it is well known that d p,w = γ p,w for any 0 < p < ∞ if and only if w satisfies B p , i.e., there exists A > 0 such that for every n ∈ N we have W p (n) ≤ AW (n) (for more details see [12,13]). Let φ be a quasiconcave sequence. The Marcinkiewicz space m φ and (resp. m 0 φ ) consists of all real sequences x = (x(n)) n∈N such that Recall that m φ and m 0 φ are symmetric spaces equipped with the norm · m φ (for more details see [9]). ments in 0 to an element in 0 and the pointwise convergence of their decreasing rearrangements. Although the similar result emerges in case of the nonatomic measure space (see [14]), the proof of it is not valid in case of the pure atomic measure space. It is worth mentioning that in the pure atomic measure space the proof of the wanted result is quite long and requires new techniques.
Assume that x m converges to x globally in measure, that is for any > 0 Then, x * m (n) converges to x * (n) for every n ∈ N.
Proof Let (x m ) ⊂ 0 , x ∈ 0 be such that x m → x globally in measure. Since for any > 0 and m ∈ N we have Without loss of generality we may assume that (b i ) is strictly decreasing. Now we present the proof in three cases.
Case 1 Suppose that card(N 1 ) = ∞. Then, it is easy to see that Now, we claim that for any n ∈ N, x * m (n) → x * (n). Indeed, by (1) we conclude that for any m ≥ M δ 1 and n ∈ N, If card(N \ N 1 ) = 0, then we are done. Otherwise, for any n ∈ N 1 and k ∈ N \ N 1 we observe that Therefore, for all m ≥ M δ 1 and n ∈ N it is easy to notice that .
In case when card (B) = j 0 then we assume that b j 0 +1 = 0. Denote for any i ∈ {1, . . . , card(B)}, Therefore, for any m ≥ M δ and Hence, for all m ≥ M δ and n i ∈ N i where 1 ≤ i ≤ j 0 − 1 we easily observe In consequence, by (3) we get for every m ≥ M δ and n ∈ N, Clearly, there exists σ : N → j 0 j=1 N j a permutation such that x * (n) = x(σ (n)) for all n ∈ N. Thus, for any n ∈ N there exists j ∈ {1, . . . , j 0 } such that σ (n) ∈ N j and by (3) we obtain (2) and (4) we infer that then without loss of generality we may assume that j 0 = card(B) and b j 0 = 0. Next, letting for any i ∈ {1, . . . , j 0 − 1}, and proceeding analogously as in case 2 we may show that x * m → x * on N, in case when card(B) < ∞. Now, assume that card(B) = ∞. Then, since (b j ) is strictly decreasing and bounded we conclude First, let us consider that b = 0. Let > 0. Then, there exists j 0 ∈ N such that for all j ≥ j 0 we have Define for any i ∈ {1, . . . , j 0 }, Similarly as in case 2 there is M δ ∈ N such that for all m ≥ M δ , n ∈ N, and k ∈ j≥ j 0 N j we get Moreover, we may observe that for every m ≥ M δ and n i ∈ N i where i ∈ {1, . . . , j 0 − 1}. Next, assuming that σ : N → ∞ j=1 N j is a permutation such that x * (n) = x(σ (n)) for all n ∈ N, then for any n ∈ N with n ≤ c j 0 −1 there exists j ∈ {1, . . . , j 0 − 1} such that σ (n) ∈ N j and by (6) we obtain On the other hand, if n > c j 0 −1 then there is j ≥ j 0 such that σ (n) ∈ N j and by (5) and (6) it follows that for all m ≥ M δ . Now, let us notice that for every n ∈ N, Hence, we infer that for any m ≥ M δ and n ∈ N, Now, we assume that b > 0. Then, it is easy to see that x * (∞) = b > 0. Next, taking for all m ∈ N, we may show that x * = y * and x * m = y * m for sufficiently large m ∈ N. Next, passing to subsequence and relabeling if necessary, it is enough to prove that y * m → y * on N. Clearly, by definition of y and y m for all m ∈ N we may observe that y m − b converges y − b globally in measure and (y − b) * (∞) = 0. Finally, using analogous technique as previously, in case 3 for b = 0, we finish the proof.
(ii) sgn(x(i)) = sgn(y(i)) for any i ∈ N and there exists (E n ) n∈N a countable collection of subsets of N such that for every n ∈ N we have card(E n ) = n and

Geometric Structure of Sequence Lorentz Spaces p,w
In this section, we discuss complete criteria for order continuity, the Fatou property, strict monotonicity and strict convexity, and also extreme points of the unit ball in the sequence Lorentz space γ p,w .
Next, passing to subsequence and relabeling if necessary we may assume that x m γ p,w ↓ d. Since W (∞) = ∞ we claim that d x (λ) < ∞ for all λ > 0 and x ∈ γ p,w . Indeed, assuming for a contrary that there is x ∈ γ p,w such that x * (∞) = lim n→∞ x * (n) > 0 we obtain ∞ → γ p,w . Define z = χ N . Then, we have z * * = z ∈ γ p,w and also z γ p,w = W (∞) = ∞, which gives us a contradiction and proves the claim. Let > 0. Define two sets Hence, by Lemma 3.1 it follows that x * m → 0 pointwise on N. Consequently, since x 1 γ p,w < ∞ and x * * (n) < ∞ for all n ∈ N, applying twice the Lebesgue Dominated Convergence Theorem we conclude x m γ p,w → 0.
Sufficiency Assume for a contrary that W (∞) < ∞. Then, it is easy to see that x = χ N ∈ γ p,w , x * * = x, and x γ p,w = W (∞). Define x m = χ {i∈N:i≥m} for any m ∈ N. Clearly, we have x m ↓ 0 and x m ≤ x pointwise for every m ∈ N. Moreover, we observe that x * * m = x * * for any m ∈ N. Hence, we get x m γ p,w = W (∞) > 0 for all n ∈ N, which contradicts with assumption that γ p,w is order continuous.

Remark 4.2
First, let us recall that for any symmetric Banach function space X over [0, ∞) with the fundamental function φ X we have X → M φ X and the embedding has norm 1, where (for more details see [2,Proposition 5.9]). Now, observe that for any symmetric Banach sequence space E over N, in [2, Proposition 5.9] is also satisfied. Namely, using analogous technique as in [2] we are able to show that for any symmetric Banach sequence space E the embedding E → m φ holds with constant 1, i.e., for all x ∈ E, we have where φ E is the fundamental sequence of E on N. Proof Let (x m ) ⊂ γ + p,w , x ∈ 0 and x m ↑ x pointwise and sup m∈N x m γ p,w < ∞. Immediately, by [2, Proposition 1.7] it follows that x * m ↑ x * . Next, applying twice Lebesgue Monotone Convergence Theorem [19] we get x m γ p,w ↑ x γ p,w . Finally, since sup m∈N x m γ p,w < ∞ it follows that x ∈ γ p,w . Proof Necessity Assume for a contrary that W (∞) < ∞. Then, we may show that ∞ → γ p,w . Next, defining two sequences x = χ {i∈N:i>1} and y = χ N we easily observe that x ≤ y, x = y, and x * * = y * * = y. Consequently, x γ p,w = y γ p,w , which contradicts with assumption that the Lorentz space γ p,w is strictly monotone.
The immediate consequence of the previous theorem and Proposition 2.1 in [10] is the following result.

Corollary 4.5 Let w ≥ 0 be a weight sequence such that W (∞) = ∞ and let 1 ≤ p < ∞. An element x ∈ S γ p,w is an extreme point of B γ p,w if and only if x * is an extreme point of B γ p,w .
Next, we show that the Lorentz space γ p,w is strictly convex for 1 < p < ∞ and w a positive weight sequence such that W (∞) = ∞. In some parts of the proof of the following theorem, we use the similar techniques to [6, Theorem 3.1] (see also [5,Theorem 2.3]). For the sake of completeness and reader's convenience we show all details of the proof. Proof Necessity Assume that γ p,w is strictly convex. For a contrary we suppose that p = 1. Let x, y ∈ S γ p,w , and x + y γ p,w = 2. Without loss of generality we may assume that x = x * and y = y * . Then, we have (x + y) * * = x * * + y * * and also x + y γ p,w = x γ p,w + y γ p,w = 2.
Consequently, since x and y are arbitrary and γ p,w is strictly convex we conclude a contradiction. Now, assume that W (∞) < ∞. Define Clearly, we have for any n ∈ N,

Moreover, we observe that
for any n ∈ N. Hence, we get Therefore, by assumption that γ p,w is strictly convex we obtain a contradiction. Now, let us suppose for a contrary that there is n 0 ∈ N such that w(n 0 ) = 0. If n 0 = 1, then take ∈ (0, 1/φ γ p,w (2)) and define It is easy to see that x = y and and also Therefore, since w(1) = 0, we have Furthermore, we observe that Hence, since w(1) = 0, we get So, in case when w(1) = 0, it follows that γ p,w is not strictly convex. Assume that n 0 > 1. Define Then, we easily observe that x = y and x γ p,w = 1. Moreover, we have Hence, since w(n 0 ) = 0, we conclude that In consequence, by assumption that γ p,w is strictly convex we get a contradiction.
Sufficiency Let x, y ∈ S γ p,w and x = y. We consider the proof in two cases.
Case 1 Assume that there exists n 0 ∈ N such that x * * (n 0 ) = y * * (n 0 ). Then, by strict convexity of the power function u p for 1 < p < ∞ we have Therefore, since for any n ∈ N, by assumption that w(n) > 0 for all n ∈ N we infer that x + y γ p,w < 2.
Indeed, assuming that it is not true it follows that (x + y) * (n) = x * (n) + y * (n) for all n ∈ N. Consequently, since W (∞) = ∞, by Remark 3.2 we obtain |x + y|(n) = |x(n)| + |y(n)| for all n ∈ N and there exists (E n ) an increasing sequence of sets such that card(E n ) = n for every n ∈ N and also i∈E n In consequence, |x(n)| = |y(n)| for any n ∈ N and so x(n) = y(n) for every n ∈ N. Therefore, in view of assumption x = y we get a contradiction. Finally, applying the triangle inequality for the maximal function we infer that Finally, we present a complete criteria for an extreme point in the ball of the Lorentz space γ 1,w . It is worth mentioning that in some parts of the proof we use similar technique to the proof in [10,Theorem 2.6]. For the sake of completeness and reader's convenience, we present all details of the proof of the following theorem. Theorem 4.7 Let w ≥ 0 be a weight sequence such that W (∞) = ∞. An element x ∈ S γ 1,w is an extreme point of B γ 1,w if and only if there exists n 0 ∈ N such that and in case when n 0 > 1, W (n 0 − 1) > 0.
Proof Letting x ∈ S γ 1,w , by Corollary 4.5 we may consider that x = x * is an extreme point of B γ 1,w . Denote Since W (∞) = ∞ and φ γ 1,w (n) = W (n) + W 1 (n) for any n ∈ N, by Lemma 4.3 and by Remark 4.2 it follows that x * (∞) = 0 and so n 0 ∈ N. We claim that x * (n 0 +1) = 0. Suppose on the contrary that x * (n 0 + 1) > 0 and denote First, notice that φ γ 1,w (n+1) > φ γ 1,w (n) > 0 for any n ∈ N. Indeed, since W (∞) = ∞ we infer that φ γ 1,w (n) > 0 for all n ∈ N. Now, assuming for a contrary that there is n ∈ N such that φ γ 1,w (n + 1) = φ γ 1,w (n), we easily obtain Hence, since w(n + 1) ≥ 0 we get a contradiction. Now, we are able to find a, b Clearly, y = z and x = (y + z)/2. Since y = y * and z = z * , by (8) we have Similarly, we may show that z γ 1,w = 1. Therefore, in view of assumption that x is an extreme point of B γ 1,w we conclude a contradiction, which proves our claim. In case when n 0 > 1 we assume that w(n) = 0 for all n ∈ {1, . . . , n 0 − 1}. Then, for a ∈ (0, x * (n 0 )) we define Next, it is clearly observe that y = z, x = (y + z)/2, y * = y = z * and Consequently, by assumption that x is an extreme point of B γ 1,w we have a contradiction. So, this implies that if n 0 > 1 then it is needed W (n 0 −1) > 0.. Now, assume that x ∈ γ 1,w and satisfies (7). For simplicity of our notation we denote c = 1/γ 1,w (n 0 ). If n 0 = 1, then by Theorem 4.4 we conclude that x is an extreme point of B γ 1,w . Consider that n 0 > 1. Suppose that y, z ∈ S γ 1,w , y = z and x = (y + z)/2. We claim that y(i) = z(i) = 0 for all i > n 0 . Indeed, if y(i) > 0 for some i > n 0 , then it is obvious that z(i) = −y(i) < 0 for some i > n 0 . Next, defining two elements On the other hand, by Theorem 4.4 we infer that u γ 1,w < y γ 1,w = 1 and v γ 1,w < z γ 1,w = 1. In consequence, we get which yields a contradiction and proves our claim. Now, define We can easily notice that y, z ∈ γ + 1,w . Indeed, if it is not true then we may define u, v ∈ γ + 1,w such that u ≤ |y|, u = |y| and v ≤ |z|, v = |z| and also x = (u + v)/2. Therefore, by Theorem 4.4 we obtain a contradiction. Next, since γ 1,w is strictly monotone and y ∈ S γ 1,w , y = x we observe that card(I 1 ) > 0 and card(I 3 ) > 0, whence y(1) > y(n 0 ). Without loss of generality we may assume that y = y * . Then, we have Moreover, by assumption that z ∈ S γ 1,w and x = (y + z)/2 it follows that z(i) = 2c − y(i) for any i ∈ {1, . . . , n 0 } and z(i) = 0 for all i > n 0 . Thus, we obtain z * (n) = (2c − y(n 0 + 1 − n)) χ {i∈N:i≤n 0 } (n) for every n ∈ N. Consequently, we have Hence, by definition of c we obtain that Furthermore, since y = y * and y(1) > y(n 0 ), we infer that for every n < n 0 , In consequence, since W (n 0 − 1) > 0, by (9) and (10) we conclude which gives us a contradiction and finishes the proof.

Application
This section is devoted to a relationship between the existence set and onecomplemented subspaces of the sequence Lorentz space γ 1,w . Next, we investigate complete criteria for the dual and predual spaces of the Lorentz space γ 1,w . Moreover, we present a complete characterization of smooth points in the sequence Lorentz space γ 1,w and its dual space and predual space. Finally, we show full criteria for extreme points in the dual space of the sequence Lorentz space γ 1,w . First, let us recall some basic definitions and notations that corresponds to the best approximation. Let X be a Banach space and C ⊂ X be a nonempty set. A continuous surjective mapping P : X → C is called a projection onto C, whenever P| C = Id, i.e., P 2 = P. Given a subspace V of a Banach space X , by P(X , V ) we denote the set of all linear bounded projections from X onto V . Let us recall that a closed subspace V of a Banach space X is said to be one-complemented if there exists a norm one projection P ∈ P(X , V ). A set C ⊂ X is said to be an existence set of the best approximation if for any x ∈ X we have It is obvious that any one-complemented subspace is an existence set. The converse in general is not true. By a deep result of Lindenstrauss [15] there exists a Banach space X and a linear subspace V of X such that V is an existence set in X and V is not one-complemented in X . However, if X is a smooth Banach space both notions are equivalent see [1,Proposition 5]). It is worth noticing that one can find in the literature concerning one-complemented subspaces a survey paper [18]. We will show that both notions are equivalent in γ 1,w , which is obviously not a smooth space.
First, we establish an identity between the sequence Lorentz spaces γ 1,w and d 1,v for some nonnegative sequences w and v.

Remark 5.1
Assuming that w = (w(n)) n∈N is a nonnegative weight sequence, we may easily observe that the identity I is a surjective isometry from the sequence Lorentz space γ 1,w onto the sequence Lorentz space d 1,v Indeed, taking x ∈ γ 1,w we observe that On the other hand, assuming that v = (v(n)) n∈N is a decreasing sequence such that lim n→∞ v(n) = 0 we may define a sequence w by w(n) = n(v(n) − v(n + 1)) for any n ∈ N and show analogously that the identity I is a surjective isometry from d 1,v onto γ 1,w .
Theorem 5.2 Let w be a nonnegative weight sequence and let V ⊂ γ 1,w , V = {0} be a linear subspace. If V is an existence set, then V is one-complemented.

Proof
Let v be a nonnegative sequence given by (11). Then, by Remark 5.1 we get that the identity I is a surjective isometry between γ 1,w and d 1,v  Now, we present the necessary and sufficient condition for the dual space of the Lorentz space γ 1,w and the isometric isomorphism between the Marcinkiewicz space and the dual space of γ 1,w . It is worth mentioning that in case of the Lorentz space d 1,w the similar result (see [14,Theorem 5.2]) was established under assumption that d 1,w is separable. x(n)y(n) for any x ∈ γ 1,w , and f γ * where y ∈ m ψ and ψ(n) = n/φ γ 1,w (n) for every n ∈ N.
Proof Sufficiency Suppose that W (∞) < ∞. We claim that ∞ → γ 1,w . Indeed, taking x = χ N it is easy to see that x * * = x and x γ 1,w = W (∞) < ∞, which implies our claim. Let f ∈ γ * 1,w . Then, by assumption there exists y ∈ m ψ such that for all n ∈ N. Next, in view of the inequality whence y ∈ 1 . Therefore, we observe that m ψ → 1 . Moreover, since γ 1,w and m ψ are symmetric by [2, Corollary 6.8] we conclude that γ 1,w → ∞ and 1 → m ψ . Hence, since ∞ is the dual space of 1 (see [16]) we have a contradiction.
Necessity Since W (∞) = ∞, by Theorem 4.1 it follows that γ 1,w is order continuous. Next, in view of Remark 5.1 the identity is a surjective isometry between γ 1,w and d 1,v where v is given by (11). Finally, by [14,Theorem 5.2] we finish the proof.
We research a complete characterization of the predual space of the Lorentz space γ 1,w . It is worth noticing that the first part of the proof of the next theorem is an immediate consequence of [ x * (n)v(n) = x γ 1,w , whence, according to (12) we finish the proof.
We investigate a full criteria for smooth points in the sequence Lorentz space γ 1,w and its dual and predual spaces. First, let us notice that by [10, Theorem 1.10] and by Remark 5.1, the next theorem follows immediately. Proof Let v be a sequence given by (11) and let V (n) = n i=1 v(i). Then, by Remark 5.1 we easily observe that V (n) = φ γ 1,w (n) = n ψ(n) for every n ∈ N and m 0 ψ = x ∈ m ψ : lim Hence, in view of [10, Theorem 1.5] we complete the proof.
Directly, by [10, Theorem 1.9] and Remark 5.1 and also Theorem 5.3 we infer the following theorem.

Remark 5.9
Although applying [10, Theorem 2.6] and Remark 5.1 we are able to find successfully an equivalent condition for an extreme point in the sequence Lorentz space γ 1,w , with w a nonnegative weight sequence, we present the proof of this problem with all details (see Theorem 4.7). It is worth mentioning that the techniques, that were presented in the proof of Theorem 4.7, might be interesting for readers and applicable to search a complete characteristic of an extreme point in γ p,w with 1 < p < ∞.