ON w-CHEBYSHEV SUBSPACES IN BANACH

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Abstract. The purpose of this paper is to introduce and discuss the con- cept of w-Chebyshev subspaces in Banach spaces. The concept of quasi. Chebyshev in ...
Bull. Korean Math. Soc. 45 (2008), No. 3, pp. 601–606

ON w-CHEBYSHEV SUBSPACES IN BANACH SPACES Maram Shams, Hamid Mazaheri, and Sayed Mansour Vaezpour

Reprinted from the Bulletin of the Korean Mathematical Society Vol. 45, No. 3, August 2008 c ⃝2008 The Korean Mathematical Society

Bull. Korean Math. Soc. 45 (2008), No. 3, pp. 601–606

ON w-CHEBYSHEV SUBSPACES IN BANACH SPACES Maram Shams, Hamid Mazaheri, and Sayed Mansour Vaezpour Abstract. The purpose of this paper is to introduce and discuss the concept of w-Chebyshev subspaces in Banach spaces. The concept of quasi Chebyshev in Banach space is defined. We show that w-Chebyshevity of subspaces are a new class in approximation theory. In this paper, also we consider orthogonality in normed spaces.

1. Introduction Let X be a real Banach space and let Y ⊂ X be a closed subspace. Consider the set-valued mapping PY : X → 2Y defined by PY (x) = {y ∈ Y : ∥x − y∥ = d(x, Y )}. We say that Y is proximinal in X if PY (x) ̸= ∅ for every x ∈ X. If Y is proximinal, PY (x) is called the best approximation set of x in Y . We say that Y is Chebyshev if PY (x) is singleton for every x ∈ X and Y is said to be quasi Chebyshev if PY (x) is nonempty and compact for every x ∈ X (see [1], [5-6]). For a Banach space X we denote its unit sphere by SX . For x ∈ X with d(x, Y ) = 1, let QY (x) = x − PY (x). It is easy to see that QY (x) = {z ∈ SX : f (z) = f (x) ∀f ∈ Y ⊥ }. For f ∈ X ∗ we define the pre-duality map of X by JX (f ) = {z ∈ SX : f (z) = ∥f ∥}. Suppose Y ⊂ X is a proximinal hyperplane and suppose f ∈ Y ⊥ , ∥f ∥ = 1 and Y = ker f . Now let x ∈ SX , d(x, Y ) = 1. Then we have |f (x)| = 1. Hence { JX (f ) if f (x) = 1, QY (x) = JX (−f ) if f (x) = −1. If we put, Yˆ = {x ∈ X : ∥x∥ = d(x, Y )} = {x ∈ X : 0 ∈ PY (x)}, then Yˆ is a closed set in X. We easily can prove that if d(x, Y ) = 1, then ∩ ∩ PY (x) = (x + (Yˆ SX )) Y. Received February 18, 2008. 2000 Mathematics Subject Classification. Primary 41A65, 46B50, 46B20, 41A50. Key words and phrases. w-Chebyshev subspaces, orthogonality, proximinal subspaces, Chebyshev subspaces. c ⃝2008 The Korean Mathematical Society

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2. w-Chebyshev subspaces In this section we define w -Chebyshev subspaces in the normed spaces and characterize them. Definition. Let X be a normed space, Y be a subspace ∩ of X. We say that W is a w -Chebyshev subspace, if for every x ∈ X, x + (Yˆ SX ) is a nonempty and weakly compact set in X. Note that if Y is w -Chebyshev, then PY (x) is weakly compact for x ∈ X\Y . Theorem 2.1. Let X be a normed space, Y be a subspace of X with codim Y = 1. Then the following statement are equivalent: i) Y is w-Chebyshev. ii) for every f ∈ Y ⊥ , JX (f ) is weakly compact. iii) for every x ∈ X, PY (x) is weakly compact. Proof. i) ⇒ ii) suppose Y is w -Chebyshev, f ∈ Y ⊥ and {zn } ⊂ JX (f ). Put ∩ h := f /∥f ∥ then we have ∥h∥ = 1, h(zn ) = ∥zn ∥, hence zn ∈ Yˆ SX . Thus the proof is complete. ii) ⇒ iii) Suppose x ∈ X and d(x, Y ) = 1. Let {zn } be a sequence in PY (x) therefore x − zn ∈ QY (x). Since for every f ∈ Y ⊥ , JX (f ) is weakly compact and in either case JY (f ) is weakly compact if QY (x) is, hence PY (x) is weakly compact. Now suppose d(x, Y ) = α, then d( α1 x, Y ) = 1. By the above prove we have PY ( α1 x) is weakly compact. Since PY ( α1 x) = α1 PY (x), the proof is complete. iii) ⇒ i) Since codim Y = 1, therefore there exists f ∈ X ∗ and Y = ker f . ∩ Let x ∈ X and {zn } be a sequence in x + (Yˆ SX ). Therefore d(zn − x, Y ) = ∥zn − x∥ = 1. ∪ On the other hand, d(zn − x, Y ) = |f (zn − x)|/∥f ∥. Hence zn − x ∈ JY (f ) JY (−f ) for each n. Thus by the above discussion, JX (f ) is weakly compact. Hence {zn − x} has a weakly convergent subsequence. ¤ We know that every quasi Chebyshev subspace with codimension one of X is a w -Chebyshev subspace of X. The following example shows that there exists a w -Chebyshev subspace of a Banach space X which is not quasi Chebyshev. 2 n ∞ Example ⊕ 2.2. Let W = l and let W0 = ⟨(−1) ⟩ be the subspace of l . Put X=W W0 and define a norm on X by,

∥x + y∥ = max{∥x∥2 , ∥y∥∞ } for all x ∈ W and all y ∈ W0 . It is clear that ∥ · ∥ is a norm on X, X is a Banach space with respect to this norm and codim W = 1. If y = (−1)n ∈ W0 , its not difficult to show that PW (y) = {x ∈ W : ∥x∥2 ≤ 1} = BW . Hence PW (y) is not compact because W is reflexive. Therefore, W is not quasi Chebyshev in X, but by Theorem 2.1, W is w -Chebyshev.

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Definition. A subspace Y of a normed space X is called w -boundedly compact if for every bounded sequence {yn } in Y , there exists x0 ∈ Y and a subsequence w {ynk } such that ynk → x0 . Note that every bounded subset of a reflexive normed space is w -boundedly compact and therefore is weakly compact. Theorem 2.3. Let Yˆ be w-boundedly compact. Then PY (x) is weakly compact. Proof. Suppose d(x, Y ) = 1 and {yn } is a sequence in QY (x). Therefore ∥yn ∥ = ∩ 1 and for every f ∈ Y ⊥ , f (yn ) = f (x). Since QY (x) ⊂ Yˆ SX and Yˆ is w w boundedly compact, there exists the subsequence {ynk } such that ynk → y0 . Hence for all f ∈ Y ⊥ , f (y0 ) = lim f (ynk ) = f (x). By Hahn Banach theorem, there exists f1 ∈ X ∗ such that ∥f1 ∥ = 1, f1 ∈ Y ⊥ and f1 (x) = 1. Since for every f ∈ X ∗ , ∥f ∥ = 1, |f (y0 )| ≤ 1 and f1 (y0 ) = 1 therefore ∥y0 ∥ = 1. Thus QY (x) is weakly compact and PY (x) is weakly compact. Now for all x ∈ X\Y let d(x, Y ) = α ̸= 0 then d( α1 x, Y ) = 1. Therefore PY ( α1 x) is weakly compact. Since PY ( α1 x) = α1 PY (x), the proof is complete. ¤ Theorem 2.4. Yˆ is w-boundedly compact if and only if Y is w-Chebyshev. Proof. Suppose Y is w -Chebyshev subspace and {yn } ⊂ Yˆ is a bounded sequence. Then there exists α such that ∥yn ∥ ≤ α. Put xn := ∥yynn ∥ , therefore ∩ xn ∈ Yˆ SX . Since Y is w -Chebyshev, there exists {xn } ⊂ {xn } such that k

w

xnk → x0 . Also {∥ynk ∥} is a bounded sequence in C then ∥yn′ ∥ → β thus w yn′ → βx0 . Also {∥ynk ∥ is a bounded sequence in the complex field C then w ∥ynK ∥ −→ β thus yn′ → βx. Then Yˆ is w -boundedly compact. ˆ ∩ SX ). Conversely, suppose Yˆ is w -boundedly compact, and {xn } ⊂ x + (W Then d(xn − x, W ) = ∥xn − x∥ = 1 therefore {xn − x} is a bounded sequence w in Yˆ and has a w -convergence sequence {xnk − x}, xnk − x → x0 . ¤ Theorem 2.5. Let M be a proximinal subspace of X and W be a subspace of X containing M . If W is a w-Chebyshev, then W/M is w-Chebyshev. Proof. We know that for every x ∈ X, d(x, W ) = d(x + M, W/M ). Suppose ˆ ) ∩ SX/M ). Then we have {yn + M } ⊆ yˆ + ((W/M d(yn − y, W ) = d((yn − y) + M, W/M ) = ∥(yn − y) + M ∥ = 1. Since M is proximinal, d(yn − y, W ) = ∥(yn − y) + mn ∥. ˆ ∩ SX . Since W is w Then mn ∈ PW (yn − y), therefore yn − y − mn ∈ W w Chebyshev there exists the subsequence {ynk −mnk } such that ynk −mnk → x0 . w Since the canonical map π : X → X/M is continuous, therefore π(ynk −mnk ) → π(x0 ). Hence for all f ∈ (X/M )∗ , f (ynk + M ) = f ◦ π(ynk ) = f ◦ π(ynk − mnk ) → f ◦ π(x0 ) = f (x0 + M ).

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Thus W/M is w-chebyshev.

Theorem 2.6. Let M be a proximinal w-boundedly compact subspace of a ˆ ) ∩ SX/M normed space X and W be a subspace of X containing M . If (W/M is compact, then W is a w -Chebyshev subspace of X. Proof. Since for all x ∈ X we have d(x, W ) = d(x + M, W/M ). Therefore ˆ ) ∩ S( X/M ). Now suppose {xn } be a sequence in x + ˆ ∩ SX ) ⊂ (W/M π(W ˆ ) ∩ SX/M where π(xn − x) = ˆ ∩ SX ) then sequence {π(xn − x)} is in (W/M (W ∩ ˆ ) SX/M is compact, there exists x0 ∈ X and a (xn − x) + M . Since (W/M subsequence such that xnk + M → x0 + M . Since M is proximinal, there exists a sequence {mnk } in M such that ∥x0 − xnk + x − mnk ∥ = d(x0 − xnk + x, M ). Therefore lim ∥x0 − xnk + x − mnk ∥ = lim ∥(xnk − x) + M − (x0 + M )∥ = 0. w

Also M is w-boundedly compact, then there exists a m0 such that mnk → m0 . Put y0 = x0 − m0 , we have |x∗ (y0 ) − x∗ (xnk − x)| = ≤

|x∗ (x0 − m0 − xnk + mnk − mnk − x)| |x∗ (x0 − xnk − x − mnk )| + |x∗ (mnk − m0 )|

≤ ∥x∗ ∥∥x0 − xnk − x − mnk ∥. w ˆ ∩ SX is closed so y0 ∈ W ˆ ∩ SX and the proof is Then xnk − x → y0 . Since W complete. ¤ Definition. let X be a Banach space. A weak∗ -closed subspace M ⊂ X ∗ is said to have property (W ∗ ) if for every x ∈ X\M⊥ , Dx = {y ∈ X : f (y) = f (x) ∀f ∈ M, ∥y∥ = ∥x∥M } is nonempty and weakly compact, where ∥x∥M = sup{|f (x)| : ∥f ∥ ≤ 1, f ∈ M }. Theorem 2.7. Let Y be w-Chebyshev subspace of X. Then Y ⊥ has the property W ∗ . Proof. let ∥x∥ = 1 and {yn } is a sequence in Dx . Therefore for all f ∈ Y ⊥ , ∩ f (yn ) = f (x) and ∥yn ∥ = ∥x∥M . Then {yn } ⊂ QY (x) ⊂ Yˆ SX , hence there w exists the subsequence {ynk } such that ynk → y0 . ¤ 3. Orthogonality in normed linear spaces Suppose X is a normed linear space and x, y ∈ X, x is said to be orthogonal to y and is denoted by x ⊥ y if and only if ∥x∥ ≤ ∥x + αy∥ for all scalar α. If M1 and M2 are subsets of X, it is defined M1 ⊥ M2 if and only if for all g1 ∈ M1 , g2 ∈ M2 , g1 ⊥ g2 . (see [3-5]). It is defined, ˘ = {x ∈ X : M ⊥x}, M the cometric set of M . At first we state lemmas which is needed in the proof of the main results.

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Lemma 3.1 (Papini and Singer [6]). Let G be a linear subspace of a normed linear space X and x ∈ X\G. The following statements are equivalent: (i) For every G ∈ g , ∥g − g0 ∥ ≤ ∥x − g∥. (ii) For each g ∈ G there exists a functional f g ∈ X ∗ such that ∥f g ∥ = 1, f g (x) = f g (g0 ) and f g (g) = ∥g∥. Theorem 3.2. Let X be a Banach space, M be a linear subspace of X and ˘ is a convex subset of X, then the following statements are F ⊆ X. If M equivalent: (i) M ⊥F . (ii) For all g ∈ M , there exists f ∈ X ∗ such that ∥f ∥ = 1 and f |F = 0. ˘ , then for each g ∈ M there exists f ∈ X ∗ such Proof. i) ⇒ ii). Since M ⊥M ˘ , it follows that f |F = 0. that ∥f ∥ = 1, f |M˘ = 0 and f (g) = ∥g∥. Now F ⊆ M ii) ⇒ i). Suppose for all g ∈ M there exists f ∈ X ∗ such that ∥f ∥ = 1, f |F = 0 and f (g) = ∥g∥. For all g ∈ M and all x ∈ F , we have ∥g∥

= f (g) = f (g + αx) ≤ ∥f ∥∥g + αx∥ = ∥g + αx∥

for all scalar α. Therefore M ⊥F .

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Corollary 3.3. Let X be a Banach space, M be a linear subspace of X and ˘ is a convex subset of X, then the following statements are x ∈ X. If M equivalent: (a) M ⊥x. (b) For all g ∈ M , ∥g∥Γx = ∥g∥, where Γx = {f ∈ X ∗ : f (x) = 0}, and ∥g∥Γx = sup |f (g)|. f ∈Γx

References [1] D. Narayana and T. S. S. R. K. Rao, Some remarks on quasi-Chebyshev subspaces, J. Math. Anal. Appl. 321 (2006), no. 1, 193–197. [2] C. Franchetti and M. Furi, Some characteristic properties of real Hilbert spaces, Rev. Roumaine Math. Pures. Appl. 17 (1972), 1045–1048. [3] H. Mazaheri and F. M. Maalek Ghaini, Quasi-orthogonality of the best approximant sets, Nonlinear Anal. 65 (2006), no. 3, 534–537. [4] H. Mazaheri and S. M. Vaezpour, Orthogonality and ϵ-orthogonality in Banach spaces, Aust. J. Math. ASnal. Appl. 2 (2005) no. 1, Art. 10, 1-5. [5] P. L. Papini and I. Singer, Best coapproximation in normed linear spaces, Monatsh. Math. 88 (1979), no. 1, 27–44. [6] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, New York-Berlin, 1970.

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Maram Shams Department of Mathematics Yazd University Yazd, Iran E-mail address: [email protected] Hamid Mazaheri Department of Mathematics Yazd University Yazd, Iran E-mail address: [email protected] Sayed Mansour Vaezpour Department of Mathematics Yazd University Yazd, Iran E-mail address: [email protected]