SQU Journal for Science, 2017, 22(1), 53-55 DOI: http://dx.doi.org/10.24200/squjs.vol22iss1pp53-55 2017 Sultan Qaboos University 53 Finite Dimensional Chebyshev Subspaces of  Aref K. Kamal Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, PC 123, Al-Khoud, Muscat, Sultanate of Oman, Email: akamal@squ.edu.om. ABSTRACT: If A is a subset of the normed linear space X, then A is said to be proximinal in X if for each xX there is a point y0A such that the distance between x and A; d(x, A) = inf{||xy||: yA}= ||xy0||. The element y0 is called a best approximation for x from A. If for each xX, the best approximation for x from A is unique then the subset A is called a Chebyshev subset of X. In this paper the author studies the existence of finite dimensional Chebyshev subspaces of   . Keywords: Best approximation; Chebyshev subspaces; Banach lattice.  في المدى محدودةالجزئية تشيبيشيف فضاءات عارف كمال بحيث ان المسافة Aفي 0yتوجد نقطة Xفي x, و كان لكل Xمجموعة جْزئية من فضاء المتجهات المعياري Aاذا كانت :ملخصال A) d(x, بين x وA | 0|تساويyx|| عندها نطلق على A " مجموعة تقريبية" في اسمX. 0النقطةy للنقطة تسمى "احسن تقريب " x منA اذا . .X" مجموعة تشيبيشيفية جزئية" من Aوحيد عندها تسمى المجموعة Aمن Xفي xكان احسن تقريب لكل في هذه الورقة يدرس المؤلف امكانية وجود فضاءات جزئية تشيبيشيفية محدودة المدى في فضاء .متشابكات باناخ و فضاءات تشيبيشيف الجزئية، احسن تقريب: مفتاحيةالكلمات ال 1. Introduction f A is a subset of the normed linear space X, then A is said to be proximinal in X if for each xX there is a point y0A such that the distance between x and A; d(x, A) = inf{||xy||: yA}= ||xy0||. In this case the element y0 is called a best approximation for x from A. If for each xX, the best approximation for x from A is unique, then the subset A is called a Chebyshev subset of X. If Q is a compact Hausdorf topological space, then C(Q) denotes the Banach space of all continuous real valued functions defined on Q equipped with the uniform norm, that is, || f || = max{| f(x) |: xQ}. For 1  p ≤ , ℓp denotes the classical Banach space of real sequences, and Lp[0, 1] denotes the classical Banach spaces of real measurable functions. Finite dimensional Chebyshev subspaces of Banach spaces have been the center of attention of mathematicians for a long time (see for example: [1-5]. One of their important properties is that the single valued metric projection function is continuous. (see, for example, [6]). In 1956 Mairhuber [7] proved a special version of what was subsequently called Mairhuber's Theorem. Mairhuber's Theorem asserts that for any compact Hausdorff space Q, and for any n ≥ 2, the Banach space C(Q) admits n dimensional Chebyshev subspaces if and only if Q is homeomorphic to a subset of a circle. ([8], Theorem 2.3, page 218). It was shown also that if Q is a compact Hausdorff space, then the n dimensional subspace N of C(Q) is a Chebyshev subspace if and only if each g ≠ 0 in N has at most n1 zeros. ([8] Theorem 2.2, page 215). In 1962, Ahiezer [9] showed that L1[0, 1] has no finite dimensional Chebyshev subspaces. It is easy to show that every finite dimensional subspace of a strictly convex space is a Chebyshev subspace ([10] page 23). Therefore for 1 < p < , every finite dimensional subspace of ℓp and every finite dimensional subspace of Lp[0, 1] is a Chebyshev subspace. In this paper the author studies the existence of the n dimensional Chebyshev subspaces of   . This is an important space of sequences, but it is not clear if it has any finite dimensional Chebyshev subspaces. In Section 2 it is shown that for n>1, this Banach space has no Chebyshev subspace of dimension n. I mailto:mohammad@squ.edu.om AREF KAMAL 54 Before ending this section some terminologies and known results, that will be used later, will be mentioned. Let ℓ denote the Banach space all real bounded sequences x = (x1, x2, …) equipped with the norm ||x||∞ = sup{|xi|: i = 1, 2, ..}. The Banach spaces X and Y are said to be isometric to each other if there is a linear mapping  from X onto Y such that ||(x)|| = ||x|| for each xX. It is clear that the isometry preserves the proximinality properties; that is, if  is an isometry from X onto Y and A is a subset of X, then for any xX, d(x, A) = d((x), (A)). Therefore x0 is a best approximation for x from A if and only if (x0) is a best approximation for (x) from (A). ( [11], page 143) shows that the space   is not separable. In Theorem 1.1 there is another proof for this fact. Theorem 1.1. : The space   is not separable. Proof: For each 0<<1, let 0.123… be the binary representation of , where i = 1or 0 for all i = 1, 2, 3, … . Define x   by x = (1, 2, 3…). The set A = {x; (0, 1)} is an uncountable subset of   , and if  ≠  then ||x  x||∞ = 1. Now let B be any dense subset of   , and let  = 1 3 , then for any  (0, 1) one must have B(x, )  B = {xB; ||x  x||< 1 3 } ≠ . For each (0, 1) choose y to be any element in B(x, )  B. It will be shown that if  ≠  in (0, 1) then y ≠ y . If this is true, then since the interval (0, 1) is uncountable and {y; (0, 1)}  B, it follows that B is uncountable. Assume that y = y for some  ≠  in (0, 1), then yB(x, )  B(x, ). But then ||x  x||∞ ≤ ||x  y||∞ + ||y  x||∞ < 1 3 + 1 3 < 1, which contradicts the fact that ||x  x||∞ = 1. Theorem 1.2. : If Q is a compact subset of the circle, then C(Q) is separable. Proof: It is clear that the set of all polynomial with rational number coefficients is a countable dense subset of C[0, 2]. So C[0, 2] is separable. Now let S be the unit circle in RR. It will be shown that C(S) is separable. If this is true, then for any compact subset Q of S, C(Q) must be separable. For each point S there is a unique [0, 2) such that  = (cos , sin ). Define : C(S)C[0, 2] by (f)() = f() if   2, and (f)(2) = (f)(0). It is clear that for each fC(S), the function (f) is continuous on [0, 2]. So  is well defined. It is also clear also that  is linear, and that ||(f)|| = ||f|| for each fC(S). So  is an isometry from C(S) into C[0, 2]. But C[0, 2] is separable, and therefore C(S) is also separable. For a proof of a more general case one can refer to ([12] Proposition 7.6.2 page 126, and Proposition 623 page 95). 2. Main Results Let X be a linear space and let ≤ be a partially ordered relation defined on X. Then (X, ≤) is said to be a lattice if for each x and y in X, the least upper bound xy and the greatest lower bound xy of x and y both exist in X. In this case if xX, then |x| is defined to be; |x| = xx. The Banach space X is called a Banach Lattice if it is a lattice and for each x and y in X, if |x| ≤ |y| then ||x|| ≤ ||y||. The element e in the Banach lattice is called a strong order unit if ||e|| = 1, and x ≤ e for all xX with ||x|| ≤ 1. The Banach lattice is called an Abstract M space if ||x+y|| = max {||x||, ||y||} for each x and y in X satisfying that xy = 0. For more information about Banach Lattices one can refer to [13]. The following theorem is Theorem 4 page 59 of [14]. Theorem 2.1. : Let X be a real Banach lattice. Then X is isometric to C(Q) for some compact Hausdorff space Q if and only if X is an abstract M space with a strong order unit. Theorem 2.2. : The Banach space   is an abstract M Banach Lattice with a strong order unit. Proof: Let ≤ be the relation defined on   such that for each (xi) and (yi) in   , (xi) ≤ (yi) if and only if xi ≤ yi for all i = 1, 2, … Then  is a partially ordered relation on   . If (xi) and (yi) are two elements in   then the least upper bound, (xi)(yi), of (xi) and (yi) is (xi)(yi) = (max{xi, yi}), and the greatest lower bound is (xi)(yi) = ( min{xi, yi}). It is clear that if (xi) and (yi) are two elements in   then both (xi)(yi) and (xi)(yi) are also elements in   , and that if | xi | ≤ | yi | for all i =1, 2, … then ||(xi)||∞ ≤ ||(yi)||∞. Therefore   with the relation ≤ is a Banach lattice. If (xi) and (yi) are in   and (xi)(yi) = 0 then min{xi, yi} = 0 for all i =1, 2, … . Therefore xi ≥ 0 and yi ≥ 0. For each i =1, 2, .. , if xi > 0 then yi = 0, and if yi > 0 then xi = 0. Thus if min {xi, yi} = 0, then xi + yi = max{xi, yi}. So for any (xi) and (yi) in FINITE DIMENSIONAL CHEBYSHEV SUBSPACES OF  55   , if (xi)(yi),= 0 then ||(xi) + (yi)||∞ = max{||(xi)||∞, ||(yi)||∞}. Thus   is an abstract M space. Finally the constant function e = (e1, e2, …) defined by ei =1 for each I = 1, 2, …, is a strong ordered unit for   . .is isometric to C(Q)  There is a compact Hausdorff space Q such that: Theorem 2.3. Proof: By Theorem 2.2,   is an abstract M Banach Lattice with a strong order unit, and by Theorem 2.1, there is a compact Hausdorff Q such that X is isometric to C(Q). The following theorem is an important theorem in Approximation Theory. Theorem 2.4. (Mairhuber's Theorem): [15]: If n >1, and C(Q) admits an n-dimensional Chebyshev subspace, then Q is homeomorphic to a subset of the circle. Theorem 2.5. : If n >1, then   has no Chebyshev subspace of dimension n. Proof: By Theorem 2.2,   is isometric to C(Q) for some compact Hausdorff space Q. If this Q is homeomorphic to a subset of the circle, then by Theorem 1.2,   is separable. But by Theorem 1.1,   is not separable. Therefore, Q is not homeomorphic to a subset of the circle. By Theorem 2.4, if n >1, then   has no n-dimensional Chebyshev subspace. 3. Conclusion If X is the Banach space   of all bounded sequences of real numbers then for n  2, X has no finite dimensional Chebyshev subspaces of dimension n. References 1. Borodin, P.A. Chebyshev subspaces in the spaces L1 and C, Mathematical Notes, 91, Issue 5(2012), 770-781. 2. Deutsch, F., Nürnberger, G., Singer, I. Weak Chebyshev subspaces and alternation, Pacific Journal of Mathematics, 1980, 89, 9-31. 3. Garkavi, A.L. “On Chebyshev and almost-Chebyshev subspaces”, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 1964, 28(4), 799-818. 4. Rakhmetov, N.K. On finite-dimensional Chebyshev subspaces of spaces with an integral matric, Mathematics of the USSR-Sbornik, 1993, 74(2), 361-382. 5. Kamal, A. On Copositive Approximation over Spaces of Continuous Functions II. The Uniqueness Of Best Copositive Approximation, Analysis in Theory and Applications, 2016, 32(1), 20-26. 6. Deutsch, F. and Lambert, J. On continuity of metric projections, Journal of Approximation Theory, 1980, 29, 116- 131. 7. Mairhuber, J.C. On Haar's theorem concerning Chebyshev approximation problems having unique solution, Proceeding of American Mathematical Society, 1956, 7, 609-615. 8. Singer, I. “Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces” Springer-Verlag, Berlin, 1970. 9. Ahiezer, N.I., Krein, M.G. “Some Questions in the Theory of Moments”, Translations of Mathematical Monographs, 2, American Mathematical. Society, Providence, R.I ,1962. 10. Cheney, E.W. “Introduction to Approximation Theory”, McGraw Hill Book Company, New York, 1966. 11. Kothe, G. “Topological Vector spaces I”, Springer Verlag 1969. 12. Semadeni, Z. “Banach spaces of continuous functions I”, PWN, Warsaw 1971. 13. Schaefer, H. “Banach Lattices and Positive operators”, Springer-Verlage, Berlin Heidelberg New York, 1974. 14. Lacey, H.E. “The Isometric Theory of Classical Banach Spaces”, Springer Verlag, 1970. 15. Schoenberg I. and Yang, C. On the unicity of solutions of problems of best approximation, Annali di Matematica Pura ed Applicata. 1961, 54, 1-12. Received 17 May 2016 Accepted 1 December 2016 http://www.sciencedirect.com/science/article/pii/0021904582900879 http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwib-sv-_frRAhXJ1hQKHTQlB_MQFggeMAE&url=http%3A%2F%2Fwww.mathnet.ru%2Fphp%2Farchive.phtml%3Fjrnid%3Dim%26wshow%3Dcontents%26option_lang%3Deng&usg=AFQjCNHFoeppuZJsKtjzrs2wLJwIWlbrow&sig2=JjAJ3hdvlN6eyy4PWb_2zw&bvm=bv.146094739,d.d24 http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0ahUKEwib-sv-_frRAhXJ1hQKHTQlB_MQFggeMAE&url=http%3A%2F%2Fwww.mathnet.ru%2Fphp%2Farchive.phtml%3Fjrnid%3Dim%26wshow%3Dcontents%26option_lang%3Deng&usg=AFQjCNHFoeppuZJsKtjzrs2wLJwIWlbrow&sig2=JjAJ3hdvlN6eyy4PWb_2zw&bvm=bv.146094739,d.d24 http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwiD9Z-PgPvRAhXDOxQKHcwJCP8QFggZMAA&url=http%3A%2F%2Fwww.global-sci.org%2Fata%2F&usg=AFQjCNHSK4UdGfB7DX_PYg-MaRQEMoI5eA&sig2=W0B3WBRO-SoYmPACUvsH8Q&bvm=bv.146094739,d.d24