Some generalizations of locally and weakly locally uniformly convex space

Nonlinear Analysis 74 (2011) 3896–3902

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Some generalizations of locally and weakly locally uniformly convex space Z.H. Zhang ∗ , C.Y. Liu College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, PR China

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Article history: Received 22 December 2009 Accepted 16 February 2011 Keywords: Strongly convex space Very convex space Almost locally uniformly rotund space Weakly almost locally uniformly rotund space k-strongly convex space k-very convex space Strongly Chebyshev set

abstract In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction In 1977, Sullivan [1] defined very convex space (also called very rotund space). Wu and Li [2] introduced and studied the strongly convex space. Many results of these two classes of convexities were discussed in [3–6]. In 2000, He [7] gave the definition of the k-strongly convex space, which is a generalization of the strongly convex space. 1-strongly convex space is equivalent to strongly convex space, and k-strongly convex space implies (k + 1)-strongly convex space. However, the converse is not generally true. Many results of k-strongly convex space were obtained in [7,8]. Recently, the author [9] discussed the important applications of the strongly convex space and the very convex space in approximation theory, and obtained some good results. In 2000, Bandgapadhyay et al. [10] proposed two generalizations of locally uniformly rotund space, which are called almost locally uniformly rotund space and weakly almost locally uniformly rotund space. Many results of these two classes of convexities were studied in [11–13]. In this paper, we will prove that almost locally uniformly rotund space is equivalent to strongly convex space and that weakly almost locally uniformly rotund space is equivalent to very convex space. Thus, we unify the results of the studies about the strongly convex space (resp. very convex space) and the almost locally uniform rotundity (resp. weakly almost locally uniform rotundity). We introduce k-very convex space, which is a generalization of very convex space, and show that 1-very convex space is equivalent to very convex space, k-very convex space implies (k + 1)-very convex space. But, the converse is not generally true. We also investigate some properties of k-strongly convex and k-very convex spaces, and show that Banach space X is strongly convex (resp. very convex) if and only if every proximinal convex set of X is norm-strongly Chebyshev (resp. weak-strongly Chebyshev).

∗

Corresponding author. Tel.: +86 21 67791195. E-mail address: [email protected] (Z.H. Zhang).

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.02.025

Z.H. Zhang, C.Y. Liu / Nonlinear Analysis 74 (2011) 3896–3902

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2. Notations and definitions Let X be a Banach space and X ∗ be its dual space. Denote S (X ) and B(X ) to be the unit sphere and the unit ball, respectively. Let x ∈ S (X ), D(x) = {x∗ ∈ S (X ∗ ) : x∗ (x) = 1}. D(S (X )) = {x∗ ∈ S (X ∗ ) : x∗ (x) = 1, ∀x ∈ S (X )}. Let x∗ ∈ S (X ∗ ), Ax∗ = {x ∈ S (X ) : x∗ (x) = 1}, F (x∗ , δ) = {x ∈ S (X ) : x∗ (x) > 1 − δ}, where 0 < δ < 1. Let {xi }ni=1 ⊆ S (X ),

  1    f 1 ( x1 ) V (x1 , . . . , xn ) = sup    . . . f (x ) n −1

fj nj=−11

Let { }

1

1 f1 (x2 )

... fn−1 (x2 )

... ... ... ...



 1  f1 (xn ) 

. . .  fn−1 (xn )

, f1 , . . . , fn−1

   ∈ B(X ∗ ) .  

∈ S (X ), ∗

  1   f1 (x1 ) n −1 n A({xi }i=1 , {fj }j=1 ) =   ...  f (x ) n−1

1

1 f1 (x2 )

... f n −1 ( x 2 )

... ... ... ...



 1  f1 (xn ) 

. . . .   fn−1 (xn )

Let M ⊆ X , and we use dim M to stand for the linear dimension of M. A Banach space X is said to be strongly convex (resp. very convex), if any x ∈ S (X ) and {xn }n∈N ⊆ B(X ) with that x∗ (xn ) → 1 as n → ∞ for some x∗ ∈ D(x) imply limn xn = x (resp. w − limn xn = x). Remark 2.1. Sullivan [1] introduced very rotund space. A Banach space X is said to be very rotund if no x∗ ∈ S (X ∗ ) is simultaneously a norming element for some x ∈ S (X ) and x∗∗ ∈ S (X ∗∗ ), where  x ̸= x∗∗ . The author [3] proved that very rotund space coincides with very convex space (also see Lemma 3.1(4)). A Banach space X is said to be almost locally uniformly rotund (ALUR) (resp. weakly almost locally uniformly rotund (wALUR)), if for any {xn }n∈N ⊆ B(X ) and {x∗m }m∈N ⊆ B(X ∗ ), the condition lim lim x∗m m



xn + x

n

2



= 1,

implies limn xn = x (resp. w − limn xn = x). ∑k+1 ∑k+1 A Banach space X is said to be k-strictly convex if for any x1 , . . . , xk+1 ∈ S (X ), the condition ‖ i=1 xi ‖ = i=1 ‖xi ‖ k+1 implies that {xi }i=1 is linearly dependent. Nan and Wang [14] proved that a Banach space X is k-strictly convex if and only if for any x∗ ∈ S (X ∗ ), dim Ax∗ ≤ k. We let dim Ax∗ = 0 if Ax∗ = ∅. Let x ∈ S (X ), x is said to be a k-smooth point of B(X ), if dim D(x) ≤ k; x is said to be a k-strongly smooth point (resp. k-very smooth point) of B(X ), if x is a k-smooth point and any {fn }n∈N ⊆ S (X ∗ ) with fn (x) → 1 as n → ∞ imply that {fn }n∈N is relatively compact (resp. relatively weakly compact). A Banach space X has one of the above properties if every point of S (X ) has the same property. Sullivan [15] introduced an important type of convex space, by the concept that k + 1 elements x1 , . . . , xk+1 encircle convex volume V (x1 , . . . , xk+1 ), which is called locally k uniformly rotund (LkUR) and which is a generalization of LUR. L1UR is equivalent to LUR. LkUR implies L(k + 1)UR, but its converses are not generally true. A natural problem is asked that: Can ALUR be generated by the concept that the k + 1 elements x1 , . . . , xk+1 encircle convex volume V (x1 , . . . , xk+1 )? Now we give a type of convex space, which is a generalization of ALUR, and it is defined by the concept that k + 1 elements x1 , . . . , xk+1 encircle convex volume V (x1 , . . . , xk+1 ). Let x ∈ S (X ), x is said to be a k-strongly convex point of B(X ), if x∗ ∈ D(x) and any {xni }n∈N ⊆ S (X ), i = 1, 2, . . . , k, with x∗ (xni ) → 1 as n → ∞ imply V (x, xn1 , . . . , xnk ) → 0 as n → ∞. A Banach space X is said to be k-strongly convex, if every point of S (X ) is a k-strongly convex point of B(X ). By [16], LkUR space implies k-strongly convex space. But the converse is not generally true. ∑k+1 ∑k+1 Let x ∈ S (X ), x is said to be a k-extreme point of B(X ), if {xi }ki=+11 ⊆ B(X ), 0 < λi < 1, i=1 λi = 1 and x = i =1 λ i x i k+1 imply that {xi }i=1 is linearly dependent. Let x ∈ S (X ), x is said to be a k-very convex point of B(X ), if x is a k-extreme point of B(X ) and any {xn }n∈N ⊆ S (X ), x∗ (xn ) → 1 as n → ∞ for some x∗ ∈ D(x) imply {xn }n∈N is relatively weakly compact. A Banach space X is said to be k-very convex, if every point of S (X ) is a k-very convex point of B(X ). Let B ⊆ X ∗ be a closed bounded convex set, a point x∗ ∈ B is said to be a weak∗ –weak point of sequential continuity (w ∗ − w seq PC) of B if {x∗n }n∈N ⊆ B and w ∗ − limn x∗n = x∗ imply w − limn x∗n = x∗ . 3. The main results Lemma 3.1. Let X be a Banach space. The following are equivalent. (1) X is k-very convex; (2) X is k-strictly convex and for any {xn }n∈N ⊆ S (X ), x∗ (xn ) → 1 as n → ∞ for some x∗ ∈ D(x) imply {xn }n∈N is relatively weakly compact;

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(3) X is k-strictly convex, and if x∗ ∈ S (X ∗ ) attains its norm on S (X ), V is a weakly open set of X and Ax∗ ⊆ V . Then there exists a δ > 0 such that x belongs to V when x is in F (x∗ , δ); (4) X is k-strictly convex, and if x∗ ∈ S (X ∗ ) attains its norm on S (X ). Then D(x∗ ) =  Ax∗ , where symbol stands for the canonical embedding of X into X ∗∗ (or X ∗ into X ∗∗∗ or X ∗∗ into X ∗∗∗∗ ). Proof. (1)→(2) By Theorem 1 in [5], if every x ∈ S (X ) is a k-extreme point, then X is k-strictly convex. (2)→(3) If (2) is not true, then there is an x∗ ∈ S (X ∗ ), a weakly open set V ⊃ Ax∗ , and a sequence {xn }n∈N ⊆ B(X ) such that xn ∈ F (x∗ , 1n ) where x∗ attains its norm on S (X ). But, {xn }n∈N ̸⊆ V . Since X is k-very convex, we know that {xn } is relatively weakly compact. Let w − limk xnk = x. Clearly, x ∈ Ax∗ ⊆ V . Hence, there exists a K such that xnk ∈ V when k > K , which is a contradiction. (3)→(4) Obviously,  Ax∗ ⊆ D(x∗ ). If (3) is not true, then there is an x∗ ∈ S (X ∗ ) and x∗∗ ∈ S (X ∗∗ ) such that x and x∗∗ attain ∗ their norm on x , that is, x∗ (x) = x∗∗ (x∗ ) = 1. But x∗∗ ̸∈ Ax∗ . Because X is k-strictly convex, then dim Ax∗ ≤ k, therefore,  Ax∗ is a weak∗ closed convex set. By the separation theorem there is an x∗0 ∈ S (X ∗ ) and ε0 > 0 such that sup x∗0 (x) + ε0 < x∗∗ (x∗0 ).

x∈Ax∗

Setting

 ε0  V = x ∈ X : x∗0 (x) < x∗∗ (x∗0 ) − . 2 Then V is a weakly open set and it includes Ax∗ . From (3), there exists a δ > 0 such that x ∈ V when x ∈ F (x∗ , δ). Since B (X ) is weakly∗ dense in B(X ∗∗ ), we have x0 ∈ B(X ) such that x∗ (x0 ) > x∗∗ (x∗ ) − δ = 1 − δ,

ε0 x∗0 (x0 ) > x∗∗ (x∗0 ) − . 2

ε

It follows that x0 ∈ F (x∗ , δ), therefore x0 ∈ V , i.e., x∗0 (x0 ) < x∗∗ (x∗0 ) − 20 , which is a contradiction. (4)→(1) By Theorem 1 in [5], if X is k-strictly convex, then every x ∈ S (X ) is a k-extreme point. Let any x ∈ S (X ) and w∗

{xn }n∈N ⊆ B(X ) with that x∗ (xn ) → 1 as n → ∞ for some x∗ ∈ D(x), denote C = { xn }n∈N . Take x∗∗ xn }, then there 0 ∈ C \ { ∗∗ ∗ ∗ ∗ exists a subnet { xn(α) }n(α)∈N of { xn }n∈N such that w − limα  xn(α) = x0 . Since w − limα x∗ (xn(α) ) = 1, we have x∗∗ 0 (x ) = 1. ∗∗  By (4), we know that x0 ∈ Ax∗ , therefore, C ⊆ X . This shows that {xn } is relatively weakly compact.  Remark 3.2. By Lemma 3.1, we can know that 1-very convex is equivalent to very convex, very convex is equivalent to very rotund, and k-very convex implies (k + 1)-very convex. But, its converse is not generally true. Example 3.3. For each k ≥ 2, there exists a k-very convex space, which is not a (k − 1)-very convex space. Let k ≥ 2 be an integer, and let i1 < i2 < · · · < ik . For each x = (α1 , α2 , . . .) ∈ l2 , we define

 ‖x ‖

2 i1 ,...,ik

=

k − j =1

2 |aij |

−

+

a2i .

i̸=i1 ,...,ik

Then Xi1 ,...,ik = (l2 , ‖ · ‖i1 ,...,ik ) is a kUR space [17], so Xi1 ,...,ik is a very convex space. But we know that Xi1 ,...,ik is not (k − 1)strictly convex by ([16], Example 1). Therefore, Xi1 ,...,ik is not a (k − 1)-very convex space. By Corollary 3.2 in [18] and Theorem 2 in [19], we have Lemma 3.4. For x ∈ S (X ), then the following are equivalent. (1) If {x∗n } ⊆ B(X ∗ ) and x∗n (x) → 1, then {x∗n }n∈N is relatively weakly compact;  (2) D (x) = D( x), where D( x) = {x∗∗∗ ∈ S (X ∗∗∗ ) : x∗∗∗ ( x) = 1, x ∈ S (X )}; (3) For every x∗ ∈ D(x), net {x∗α } ⊆ B(X ∗ ), w ∗ − limα x∗α = x∗ implies w − limα x∗α = x∗ . By Lemma 3.4, we can easily get

 Lemma 3.5. A Banach space X is k-very smooth if and only if for any x ∈ S (X ), D (x) = D( x), and dim D(x) ≤ k. Theorem 3.6. A Banach space X is very convex if and only if every x∗ ∈ D(S (X )) is a smooth point of B(X ∗ ). Proof. Necessity. If there exists an x0 ∈ S (X ), and x∗0 ∈ D(x0 ) is not a smooth point of B(X ∗ ). Then dim D(x∗0 ) > 1. Since X is very convex, by Lemma 3.1(4), we get D(x∗0 ) =  Ax∗ . It follows that dim  Ax∗ > 1, therefore, X is not strictly convex. This is a 0 0 contradiction for the assumption of necessity. ∗ Sufficiency. Suppose x ∈ S (X ) and {xn } ⊆ B(X ) with that x (xn ) → 1 for some x∗ ∈ D(x). Since x∗ is a smooth point of B(X ∗ ), we have w ∗ − limn xn =  x, thus w − limα xn = x. It follows that X is very convex. Bandgapadhyay et al. (Proposition 4.4 in [11]) proved that: x is a wALUR point of B(X ) if and only if every x∗ ∈ D(x) is a smooth point of B(X ∗ ). Hence we may know that : X is wALUR if and only if for any x∗ ∈ D(S (X )) is a smooth point of B(X ∗ ). By this conclusion and Theorem 3.6, we obtain the following theorem. 

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Theorem 3.7. A Banach space X is very convex if and only if it is wALUR. Bandgapadhyay et al. (Corollary 6.9 in [11]) proved that: x is a wALUR point of B(X ) as well as w ∗ − w seq PC of B(X ∗∗ ) if and only if every x∗ ∈ D(x) is a very smooth point of B(X ∗ ). Hence, we may know that: a Banach space X is wALUR and any x ∈ S (X ) is w ∗ − w seq PC of B(X ∗∗ ) if and only if any x∗ ∈ D(S (X )) is a very smooth point of B(X ∗ ). Hence, we can obtain Theorem 3.8. A Banach space X is very convex and any x ∈ S (X ) is w ∗ − w seq PC of B(X ∗∗ ) if and only if any x∗ ∈ D(S (X )) is a very smooth point of B(X ∗ ). Under the continuum hypothesis, Nan and Wang [14] proved the following lemma. Lemma 3.9. Let x ∈ S (X ) be a k-smooth point. If {x∗n }n∈N ⊂ S (X ∗ ) and x∗n (x) → 1 as n → ∞, then {x∗n }n∈N has a weakly ∗ convergent subsequence. Under continuum hypothesis, using Lemmas 3.1, 3.5 and 3.9, we can prove the following theorem, which is a generalization of Theorem 3.8. Theorem 3.10. A Banach space X is k-very convex and any x ∈ S (X ) is w ∗ − w seq PC of B(X ∗∗ ) if and only if any x∗ ∈ D(S (X )) is a k-very smooth point. Proof. Necessity. If there exists an x0 ∈ S (X ) and x∗ ∈ D(x0 ) such that x∗0 is not a k-very smooth point of B(X ∗ ). From Lemma 3.5, we have three cases as follows.  Case I. D (x∗ ) = D(x∗ ), but dim D(x∗ ) > k 0

0

0

Since X is k-very convex, by Lemma 3.1(4), we have that  Ax∗ = D(x∗0 ). This means that dim  Ax∗ > k. Hence, X is not 0 0 k-strict convex. This is a contradiction for the assumption of necessity.

 Case II. dim D(x∗0 ) ≤ k, but D (x∗0 ) ̸= D(x∗0 ) ∗ ∗  Clearly, D (x ) ⊆ D(x0 ). Arbitrarily select x∗∗∗∗ ∈ D(x∗0 ), then x∗∗∗∗ ∈ S (X ∗∗∗∗ ) and x∗∗∗∗ (x∗0 ) = 1. Since X ∗∗∗ = 0  ∗ ⊥ 0 ∗∗ ∗∗∗∗ ∗∗ ∗ ⊥ ∗∗ ∗∗ ∗ ⊥ ∗ ⊥ ∗ ⊥ ∗∗∗∗  X (X ) , we have x = x + (x ) , where  x ∈  X , (x ) ∈ (X ) , (X ) = {x ∈ X ∗∗∗∗ : x∗∗∗∗ ( x∗ ) = 0, for all x∗ ∈ X ∗ }. Because x∗∗∗∗ (x∗0 ) = 1, by the decomposition theorem above, we get x∗∗ (x∗0 ) = 1, that is, x∗ ∈ D(x∗0 ). In 0 ∗∗ view of Goldstine theorem, we can choose a net {x∗∗ α } ⊆ B(X ) such that

w ∗ − lim x∗α = x∗∗∗∗ =  x∗∗ + (x∗ )⊥ . α

Because ∗ x∗∗ x∗∗ + (x∗ )⊥ )(x∗0 ) = 1, α (x0 ) → (

thus ‖x∗α ‖ → 1, therefore, we may assume that ‖x∗α ‖ = 1. Take α1 ≺ α2 ≺ · · · ≺ αn ≺ · · ·, such that ∗ |x∗∗ αn (x0 ) − 1| <

1 n

.

∗ (i) If there exists α0 ≻ αn , n = 1, 2, . . . , then x∗∗ α ∈ D(x0 ) where α ≻ α0 . Since X is k-very convex, by Lemma 3.1(4), we  can know that D(x∗ ) =  A ∗ . Therefore, by Lemma 3.1(2), we know that  A ∗ is weakly compact. Further, D (x∗ ) is weakly∗ 0

x0

x0

0

   x∗ ) ⊆ D (x∗0 ). compact. It follows that x∗∗ +(x∗ )⊥ ∈ D (x∗0 ), thus (x∗ )⊥ = 0. This means that x∗∗∗∗ =  x∗∗ ∈ D (x∗0 ), that is, D( ∗ ∗∗ (ii) If there exists n = n(α) satisfying αn ≻ α , then {x∗∗ } is a cofinal subsequence of the net { x } . Since dim D ( x ) α∈ N αn α 0 ≤ k, ∗ ∗∗ ∗ thus x∗0 is a k-smooth point of B(X ∗ ). Since x∗∗ αn (x0 ) → 1 as n → ∞, by Lemma 3.9, we know that {xαn }n∈N has a weak ∗ ∗ ∗∗ ∗∗ ∗∗ = y , clearly, y ∈ D ( x ) convergent subsequence. For convenience we also write it as {x∗∗ } . Let w − lim x n αn αn n∈N 0 .

Since X is very convex, by Lemma 3.1(4), we know that D(x∗0 ) =  Ax∗ . This shows that there exists a  y ∈ Ax∗ such that 0 0 ∗ ∗∗ ∗∗ ∗ ∗∗     y = y. From the assumption w − limn xαn = y ∈ D(x0 ), we can get that w − limn xαn = y. Based on the uniqueness    of a weak∗ limit, we know that x∗∗∗∗ =  y∗∗ =  y∈D (x∗ ). It follows that D(x∗ ) ⊆ D (x∗ ). 0

0

0

 Combining the two cases of (i) and (ii), we can conclude that D(x∗0 ) = D (x∗0 ), which is a contradiction.  Case III. D (x∗ ) ̸= D(x∗ ), but dim D(x∗ ) > k 0

0

0

The same as the proof of case I, we can prove case III is impossible. Sufficiency. If X is not k-strictly convex, then there exists an x∗0 ∈ S (X ∗ ) such that dim Ax∗ > k. Clearly,  Ax∗ ⊆ D(x∗0 ), 0 0 ∗ therefore, dimD(x0 ) > k. This is a contradiction for the assumption of sufficiency. Suppose that x ∈ S (X ), {xn }n∈N ⊆ B(X ) ∗ ∗ ∗ ∗ with x (xn ) → 1 as n → ∞ for some x ∈ D(x). Since x is a k-very smooth point of B(X ), we know that { xn } is relatively weakly compact. Let w − limk  xn = x∗∗ . Since B (X ) is a weakly closed convex subset of X ∗∗ , we have that {xn }n∈N is relatively k

weakly compact. By Lemma 3.1(2), we know that X is k-very convex. Since x∗ ∈ D(S (X )) is a very smooth point, by Lemma 3.4, we know that any x ∈ S (X ) is a w ∗ − w seq PC of B(X ∗∗ ).



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Theorem 3.11. Let X be a Banach space. Then (1) If X is a k-very convex space. Then for each x ∈ S (X ), f ∈ D(x), and any {xni }ki=1 ⊆ S (X ) with f (xni ) → 1 as n → ∞ imply A({x} ∪ {xni }ki=1 , {fj }kj=1 ) → 0 as n → ∞ for any {fj }kj=1 in S (X ∗ ). (2) If each x ∈ S (X ), f ∈ D(x) with Af = k, and any {xni }ki=1 ⊆ S (X ) with f (xni ) → 1 as n → ∞ imply A({x}∪{xni }ki=1 , {fj }kj=1 ) → 0 as n → ∞ for any {fj }kj=1 in S (X ∗ ). Then X is a k-very convex space.

Proof. (1) If there exist {xni }n∈N in S (X ), and some f ∈ D(x) with f (xni ) → 1 as n → ∞, i = 1, 2, . . . , k. But, for fix {fj0 }kj=1 in S (X ∗ ), limn A({x} ∪ {xni }ki=1 , {fj0 }kj=1 ) ̸= 0. Then there exists a subsequence of {n}, which is also written as {n},

such that limn A({x} ∪ {xni }ki=1 , {fj0 }kj=1 ) exists and it is unequal to zero. Since f (xni ) → 1 as n → ∞, i = 1, 2, . . . , k, according to the assumption, we know that {xni }n∈N , i = 1, 2, . . . , k, are all relatively weakly compact. Therefore, we may select a subsequence {m} of {n} by diagonal method such that {xm i }, i = 1, 2, . . . , k, all are weakly convergent. Let w− limm xm i = xi , i = 1, 2, . . . , k. Obviously, xi ∈ Af . Since X is k-strictly convex, thus dim Af ≤ k, it follows that x, x1 , . . . , xk are linearly dependent. Without loss of generality, let x =

  f (x) f (x1 )   f (x) f1 (x1 ) k k A({x} ∪ {xi }i=1 , {fj }j=1 ) =  1 ...  ...  f (x) f (x ) k k 1    k −  f αi xi    i=1   k −   f1 αi xi =  i =1    ...   k −   fk αi xi 

... ... ... ...

∑k

i=1

αi xi . Then for any {fj }kj=1 , we have that

f (xk )   f1 (xk )



. . .  fk (xk )

f (x1 )

...

f1 (x1 )

...

...

...

fk (x1 )

...

i =1

  f (xk )     f1 (xk )   ...    fk (xk )

= 0. Hence k k k k lim A({x} ∪ {xm i }i=1 , {fj }j=1 ) = A({x} ∪ {xi }i=1 , {fj }j=1 ) = 0. m

But, since limn A({x} ∪ {xni }ki=1 , {fj0 }kj=1 ) exists and is unequal to zero, we have that k 0 k lim A({x} ∪ {xm i }i=1 , {fj }j=1 ) ̸= 0, m

which is a contradiction. (2) If X is not k-strictly convex, then there exists an x∗0 ∈ S (X ∗ ) such that dimAx∗ > k, so there are at least (k + 1) linearly 0 independent elements x0 , x1 , . . . , xk ∈ Ax∗ . Clearly, x∗0 (xi ) = 1, i = 0, 1, . . . , k. By Hahn–Banach Theorem, we choose 0 fi ∈ S (X ∗ ), i = 1, . . . , k, such that fi (xj ) =



bi > 0, 0,

i=j , i ̸= j

i, j = 1, 2, . . . , k

hence

  1   f1 (x0 )  ...   f (x ) k

0

1 f1 (x1 )

... fk (x1 )

... ... ... ...



1   f1 (xk )

. . .  fk (xk )

= b1 b2 , . . . , bk > 0.

This is contraction for the assumption. Arbitrarily take {xn }n∈N ⊆ B(X ) with f (xn ) → 1 as n → ∞ for some f ∈ D(x). We will prove that {xn }n∈N is relatively weakly compact. Let C be a set of weak∗ cluster point of {xn }n∈N . Arbitrarily take x∗∗ xn }n∈N , then there 0 ∈ C \ { ∗∗ exists a subnet { xn(α) }n(α)∈N of {xn }n∈N such that w ∗ − limα  xn(α) = x∗∗ . Therefore, we have that lim x α n(α) (fj ) = x0 (fj ) 0 k−1 k ∗ for any {fj }j=1 in S (X ). Since dim Af = k, we can choose {xi }i=1 in Af such that x, x1 , . . . , xk−1 are linearly independent. Let xn1 = x1 , . . . , xnk−1 = xk−1 , then f (xni ) = 1, i = 1, 2, . . . , k − 1. We notice that limn f (xn ) = 1, we have that limα f (xn(α) ) = x∗∗ 0 (f ) = 1. Since every subnet of a convergent net converges to same limit as the net, according to the assumption, we obtain that n(α) k−1 i =1

lim A({x} ∪ {xn(α) } ∪ {xi α

}

, {fj }kj=1 ) = lim A({x} ∪ {xn } ∪ {xni }ki=−11 , {fj }kj=1 ) = 0, n

Z.H. Zhang, C.Y. Liu / Nonlinear Analysis 74 (2011) 3896–3902

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for any {fj }kj=1 in S (X ∗ ). Hence n(α) k−1 i =1

k−1 k A({x} ∪ {x∗∗ 0 } ∪ {xi }i=1 , {fj }j=1 ) = lim A({x} ∪ {xn(α) } ∪ {xi

α

}

, {fj }kj=1 ) = 0.

This shows that x, x1 , . . . , xk−1 , x∗∗ 0 are linearly dependent. Because x, x1 , . . . , xk−1 are linearly independent, we get that x∗∗  0 ∈ span{x, x1 , . . . , xk−1 } ⊆ X . It follows that {xn } is relatively weakly compact. The author [8] proved the following lemma. Lemma 3.12. Let X be a Banach space. Then the following are equivalent. (1) X is k-strongly convex; (2) Every f ∈ D(S (X )) is a k-strongly smooth point of B(X ∗ ); ∗ (3) X is k-strictly convex, and for any x ∈ S (X ) and any f ∈ D(x), the duality mapping D : S (X ∗ ) → 2S (X ) is norm–norm upper semi-continuous at f and D(f ) is a compact set; (4) If X is k-strictly convex, any x ∈ S (X ), f ∈ D(x), F ∈ D(f ) and net {Fα }, w ∗ − limα Fα = F , then w − limα Fα = F ; (5) If X is k-strictly convex, any x ∈ S (X ), {xn }n∈N ⊆ B(X ) and f (xn ) → 1 as n → ∞ for some f ∈ D(x), then {xn }n∈N is a relatively compact set. Remark 3.13. (1) By Theorem 3.11, we can know that 1-strong convex space is equivalent to strongly convex space, and k-strongly convex space implies (k + 1)-strongly convex space. But the converse is not generally true by Example 3.3. (2) By Lemma 3.1(2) and Lemma 3.12(2), we may know that k-strongly convex space implies k-very convex space. But, the converse is not generally true. Example 3.14. For each k ≥ 2, there exists a k-very convex space, which is not a k-strongly convex space. Let X be a Hilbert space, and {ei }∞ i=0 be an orthonormal basis. For any x ∈ X , x = λ0 e0 + λ1 e1 + · · · + λn en + · · ·, let

‖x‖21 = max{|λ20 | + |λ22 | + · · · + |λ22n | + · · · , |λ21 | + |λ23 | + · · · + |λ22n+1 | + · · ·}. It is obvious that ‖ · ‖1 is an equivalent norm on X . Further, we set

 2 1

‖x ‖ = ‖x ‖ +

k+1 − 1 i =1

2i

 21 2 i

|λ |

.

Clearly, this is again a norm on X which is an equivalent original one. From Example 2.8.27 in [20], we know that (X , ‖ · ‖) is strict convex, reflexive and does not have the property (H). Hence, by Lemma 3.1(2), we know that (X , ‖ · ‖) is a k-very convex space. By Theorem 4 in [6], we know that k-strongly convex implies property (H), so (X , ‖ · ‖) is not a k-strongly convex space. and Corollary 3.15. A Banach space X is strongly convex if and only if every f ∈ D(S (X )) is a strongly smooth point of B(X ∗ ). Bandgapadhyay et al. (Corollary 4.6 in [11]) proved that: x is a ALUR point of B(X ) if and only if every x∗ ∈ D(x) is a Fréchat smooth point of B(X ∗ ). Hence we may know that: X is ALUR if and only if every x∗ ∈ D(S (X )) is a Fréchat smooth point of B(X ∗ ). By this conclusion and Corollary 3.15, we can obtain the following theorem. Theorem 3.16. A Banach space X is strongly convex if and only if it is ALUR. Recall that x∗ ∈ S (X ∗ ) is called a norm-strongly exposing (weak-strongly exposing) functional if x∗ ∈ NA(X ) and every {xn } ⊆ B(X ) with limn x∗ (xn ) = 1 is convergent, where NA(X ) stands for the set of all norm-attaining functions in X ∗ (see [13]). Recently, Bandyopadhyay et al. [13] discussed the relationship between ALUR (resp. wALUR) and norm-strongly Chebyshev (resp. weak-strongly Chebyshev) set of X . By Theorem 11 in [13], we can derive the following results, which show that strongly convex space and very convex space have important applications in approximation theory. Theorem 3.17. For a Banach space X , the following are equivalent. (1) (2) (3) (4) (5)

X is a strongly convex space (resp. very convex space); Every x∗ ∈ NA(X ) is a norm-strong exposing functional (resp. weak-strong exposing functional); Every proximinal convex set B ⊆ X is norm-strongly Chebyshev (resp. weak-strongly Chebyshev); Every proximinal subspace B ⊆ X is norm-strongly Chebyshev (resp. weak-strongly Chebyshev); Every proximinal hyperplane B ⊆ X is norm-strongly Chebyshev (resp. weak-strongly Chebyshev). By Theorem 2.5 in [13], we have

Corollary 3.18. Let X be a Banach space and B be a proximinal convex set in X , we have (1) If X is a strongly convex space, then the metric projection PB is single-valued and norm–norm continuous. (2) If X is very convex space, then the metric projection PB is single-valued and norm-weak continuous.

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