Best approximation in spaces with asymmetric norm

Costica Mustata
(original), Approximation Theory, asymmetric norm, best approximation, paper

Abstract


In this paper we shall present some results on spaces with asymmetric seminorms, with emphasis on best approximation problems in such spaces.

Authors

Stefan Cobzas
Babes-Bolyai University, Cluj-Napoca, Romania

Costică Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy,  Romania

Keywords

Spaces with asymmetric norm; best approximation; Hahn-Banach theorem; characterization of best approximation.

Paper coordinates

S. Cobzas, C. Mustăţa, Best approximation in spaces with asymmetric norm, Rev. Anal. Numer. Theor. Approx. 35 (2006) no. 1, 17-31.

PDF

https://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no1-art4/Cobzas-Mustata-pdf

About this paper

Journal

Revue d’Analyse Numer. Theor. Approx.

Publisher Name

Publishing House of the Romanian Academy

DOI

https://ictp.acad.ro/jnaat/journal/article/view/2006-vol35-no1-art4

Print ISSN

2501-059X

Online ISSN

2457-6794

google scholar link

[1] Alegre, C.Ferrer, J. andGregori, V.,Quasi-uniformities on real vector spaces,Indian J. Pure Appl. Math., 28, no. 7, pp. 929–937, 1997.
[2]___,On the Hahn-Banach theorem in certain linear quasi-uniform structures, ActaMath. Hungar.,82, no. 4, pp. 325–330, 1999.
[3] Alimov, A. R.,The Banach-Mazur theorem for spaces with nonsymmetric distance,Uspekhi Mat. Nauk, 58, no. 2, pp. 159–160, 2003.
[4] Babenko, V. F.,Nonsymmetric approximations in spaces of summable functions,Ukrain. Mat. Zh., 34, no. 4, pp. 409–416, 538, 1982.
[5]___,Nonsymmetric extremal problems of approximation theory, Dokl. Akad. NaukSSSR,269, no. 3, pp. 521–524, 1983.
[6]___,Duality theorems for certain problems of the theory of approximation, Currentproblems in real and complex analysis, Akad. Nauk Ukrain. SSR Inst. Mat., Kiev,pp. 3–13, 148, 1984.
[7] Borodin, P. A.,The Banach-Mazur theorem for spaces with an asymmetric norm andits applications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.
[8] Cobzas, S.,Phelps type duality results in best approximation, Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.
[9]___,Separation of convex sets and best approximation in spaces with asymmetricnorm, Quaest. Math.,27, no. 3, 275–296, 2004.
[10]___,Asymmetric locally convex spaces,Int. J. Math. Math. Sci., no. 16, 2585–2608,2005.
[11] Cobzas, S. and Mustata, C.,Extension of bilinear functionals and best approximationin 2-normed spaces, Studia Univ. Babes-Bolyai, Mathematica, 43, pp. 1–13, 1998.
[12]___,Extension of bounded linear functionals and best approximation in spaces withasymmetric norm, Rev. Anal. Numer. Theor. Approx.,33, no. 1, pp. 39–50, 2004.
[13] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx.Theory,94, no. 1, pp. 54–72, 1998.
[14] Dolzhenko, E. P. andSevastyanov, E. A.,Approximations with a sign-sensitiveweight (existence and uniqueness theorems), Izv. Ross. Akad. Nauk Ser. Mat.,62, no. 6,pp. 59–102,1998.
[15]___,Sign-sensitive approximations, J. Math. Sci. (New York),91, no. 5, pp. 3205–3257, 1998.
[16]___,Approximation with a sign-sensitive weight (stability, applications to snake the-ory and Hausdorff approximations), Izv. Ross. Akad. Nauk Ser. Mat.,63, no. 3, pp.77–118, 1999.
[17] Ferrer, J.,Gregori, V. andAlegre, C.,Quasi-uniform structures in linear lattices,Rocky Mountain J. Math.,23, no. 3, pp. 877–884, 1993.
[18] Garcıa-Raffi, L. M.,Romaguera, S. and Sanchez Perez, E. A.,Extensions of asymmetric norms to linear spaces, Rend. Istit. Mat. Univ. Trieste,33, nos. 1–2, 113–125, 2001.
[19]___,The bicompletion of an asymmetric normed linear space, Acta Math. Hungar.,97, no. 3, pp. 183–191, 2002.
[20]___,Sequence spaces and asymmetric norms in the theory of computational com-plexity, Math. Comput. Modelling,36, nos. 1–2, pp. 1–11, 2002.
[21]___,The dual space of an asymmetric normed linear space, Quaest. Math.,26, no. 1,pp. 83–96, 2003.
[22]___,On Hausdorff asymmetric normed linear spaces, Houston J. Math.,29, no. 3,pp. 717–728 (electronic), 2003.
[23] Krein, M. G. and Nudelman, A. A., The Markov Moment Problem and ExtremumProblems, Nauka, Moscow, 1973 (in Russian), English translation: American Mathe-matical Society, Providence, R.I., 1977.
[24] Chong Li, On wel l posed generalized best approximation problems, J. Approx. Theory,107, no. 1, pp. 96–108, 2000.
[25] Chong Liand Renxing Ni, Derivatives of generalized distance functions and existenceof generalized nearest points, J. Approx. Theory,115, no. 1, pp. 44–55, 2002.
[26] Mohebi, H.,On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory,107,no. 1, pp. 87–95, 2000.
[27]___,Pseudo-Chebyshev subspaces inL1, Korean J. Comput. Appl. Math.,7, no. 2,pp. 465–475, 2000.
[28]___,On pseudo-Chebyshev subspaces in normed linear spaces, Math. Sci. Res. Hot-Line,5, no. 9, pp. 29–45, 2001.
[29]___,Quasi-Chebyshev subspaces in dual spaces, J. Nat. Geom.,20, nos. 1-2, pp.33–44, 2001.
[30]___,On pseudo-Chebyshev subspaces in normed linear spaces, J. Nat. Geom.,24,nos. 1–2, pp. 37–56, 2003.
[31] Mohebi, H. andRezapour,Sh.,On weak compactness of the set of extensions of acontinuous linear functional, J. Nat. Geom.,22, nos. 1–2, pp. 91–102, 2002.
[32] Mustata, C.,Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[33]___,On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx.,31, no. 1, pp. 103–108, 2002.
[34]___,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ.Baia Mare, Ser. B, Matematica-Informatica,18, no. 2, pp. 275–280, 2002.
[35]___,On the uniqueness of the extension and unique best approximation in the dual ofan asymmetric linear space, Rev. Anal. Numer. Theor. Approx.,32, no. 2, pp. 187–192,2003.

[36] Renxing Ni,Existence of generalized nearest points, Taiwanese J. Math.,7, no. 1, pp.115–128, 2003.
[37] Phelps, R. R.,Uniqueness of Hahn-Banach extensions and best approximations, Trans.Amer. Marth Soc.,95, pp. 238–255, 1960.
[38] Rezapour, Sh.,∈-weakly Chebyshev subspaces of Banach spaces, Anal. Theory Appl.,19, no. 2, pp. 130–135, 2003.
[39] Romaguera, S. and Sanchis, M.,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103, no. 2, pp. 292–301, 2000.
[40] Romaguera, S. and Schellekens, M.,Duality and quasi-normability for complexityspaces, Appl. Gen. Topol.,3, no. 1, pp. 91–112, 2002.
[41] Simonov, B. V.,On the element of best approximation in spaces with nonsymmetricquasinorm, Mat. Zametki,74, no. 6, pp. 902–912, 2003.
[42] Singer, I.,Best Approximation in Normed Linear Spaces by Elements of Linear Sub-spaces, Editura Academiei Romane and Springer-Verlag, Bucharest-New York-Berlin,1970.
[43] Zanco, C. and Zucchi, A.,Moduli of rotundity and smoothness for convex bodies,Bolletino U. M. I., (7),7-B, pp. 833–855, 1993

Paper (preprint) in HTML form

BEST APPROXIMATION IN SPACES WITH ASYMMETRIC NORM S. COBZAS¸ * and C. MUST ˘ AT¸A † Abstract. In this paper we shall present some results on spaces with asymmet- ric seminorms, with emphasis on best approximation problems in such spaces. MSC 2000. 41A65. Keywords. Spaces with asymmetric norm, best approximation, Hahn-Banach theorem, characterization of best approximation. 1. INTRODUCTION Let X be a real vector space. An asymmetric seminorm on X is a positive sublinear functional p : X → [0, ∞), i.e. p satisfies the conditions: (AN1) p(x) ≥ 0; (AN2) p(tx)= tp(x); (AN3) p(x + y) ≤ p(x)+ p(y), for all x, y ∈ X and t ≥ 0. The function ¯ p : X → [0, ∞) defined by ¯ p(x)= p(-x),x ∈ X, is another positive sublinear functional on X , called the conjugate of p, and (1.1) p s (x) = max{p(x),p(-x)},x ∈ X, is a seminorm on X and the inequalities (1.2) |p(x) - p(y)|≤ p s (x - y) and | ¯ p(x) - ¯ p(y)|≤ p s (x - y) hold for all x, y ∈ X. If the seminorm p s is a norm on X then we say that p is an asymmetric norm on X . This means that, beside (AN1)–(AN3), it satisfies also the condition (AN4) p(x)=0 and p(-x) = 0 imply x =0. The pair (X, p), where X is a linear space and p is an asymmetric seminorm on X is called a space with asymmetric seminorm, respectively a space with asymmetric norm, if p is an asymmetric norm. * “Babe¸s-Bolyai” University, Faculty of Mathematics and Computer Science, 400084 Cluj-Napoca, Romania, e-mail: scobzas@math.ubbcluj.ro. † “T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania, e-mail: cmustata@ictp.acad.ro.
The function ρ : X × X → [0; ∞) defined by ρ(x, y)= p(y - x), x,y ∈ X, is an asymmetric semimetric on X . Denote by B p (x, r)= {x ∈ X : p(x - x) <r} and B p (x, r)= {x ∈ X : p(x - x) ≤ r}, the open, respectively closed, ball in X of center x and radius r> 0. Denoting by B p = B p (0, 1) and B p = B p (0, 1), the corresponding unit balls then B p (x, r)= x + rB p and B p (x, r)= x + rB p . The unit balls B p and B p are convex absorbing subsets of the space X and p agrees with the Minkowski functional associated to any of them. Recall that for an absorbing subset C of X the Minkowski functional p C : X → [0; ∞) is defined by p C (x) = inf {t> 0: x ∈ tC }. If C is absorbing and convex, then p C is a positive sublinear functional, and {x ∈ X : p C (x) < 1}⊂ C ⊂{x ∈ X : p C (x) ≤ 1}. An asymmetric seminorm p generates a topology τ p on X , having as basis of neighborhoods of a point x ∈ X the family {B p (x, r): r> 0} of open p-balls. The family {B p (x, r): r> 0} of closed p-balls is also a neighborhood basis at x for τ p . The topology τ p is translation invariant, i.e. the addition + : X × X → X is continuous, but the multiplication by scalars · : R × X → X need not be continuous, as it is shown by some examples (see [7]). The ball B p (x, r) is τ p -open but the ball B p (x, r) need not to be τ p -closed, as can be seen from the following typical example. Example 1.1. Consider on R the asymmetric seminorm u(α) = max{α, 0}, α ∈ R, and denote by R u the space R equipped with the topology τ u gener- ated by u. The conjugate seminorm is ¯ u(α)= - min{α, 0}, and u s (α)= max{u(α), ¯ u(α)} = |α|. The topology τ u , called the upper topology of R, is generated by the intervals of the form (-∞; a), a ∈ R, and the family {(-∞; α + ): > 0} is a neighborhood basis of every point α ∈ R. The set (-∞; 1) = B u (0, 1) is τ u -open, and the ball B u (0, 1) = (-∞; 1] is not τ u -closed because R \ B u (0, 1) = (1; ∞) is not τ u -open. The topology τ p could not be Hausdorff even if p is an asymmetric norm on X . A necessary and sufficient condition in order that τ p be Hausdorff was given in [22]. Putting (1.3) ˜ p(x) = inf {p(x )+ p(x - x): x ∈ X },x ∈ X, it follows that ˜ p is the greatest (symmetric) seminorm majorized by p and the topology τ p is Hausdorff if and only if ˜ p(x) > 0 for every x =0. Changing x
to -x and taking x = 0 it follows that, in this case, p(x) > 0 for every x =0, but this condition is not sufficient for τ p to be Hausdorff, see [22]. Spaces with asymmetric seminorms were investigated in a series of papers, emphasizing similarities with seminormed spaces as well as differences, see [1, 2, 3, 7, 17, 18, 19, 21, 22], and the references quoted therein. Among the differences we mention the fact that the dual of a space with asymmetric semi- norm is not a linear space but merely a convex cone in the algebraic dual X # of X. This is due to the fact that the continuity of a linear functional ϕ on (X, p) does not imply the continuity of -ϕ. For instance, ϕ(u)= u is contin- uous on (R,u) but ψ(u)= -u is not continuous. For an other example see [7]. The study of spaces with asymmetric norm was motivated and stimulated also by their applications in the complexity of algorithms, see [20, 40]. Some continuity properties of linear functionals in the symmetric case have their analogs in the asymmetric one. Proposition 1.2. [21] Let (X, p) be a space with asymmetric seminorm and ϕ : X → R a linear functional. Then the following are equivalent. (1) ϕ is τ p -τ u -continuous at 0 ∈ X . (2) ϕ is τ p -τ u -continuous on X . (3) There exists L ≥ 0 such that (1.4) ∀x ∈ X, ϕ(x) ≤ Lp(x). (4) ϕ is upper semi-continuous from (X, τ p ) to (R, ||). A linear functional satisfying (1.4) is called semi-Lipschitz (or p-bounded) and L a semi-Lipschitz constant. Denote by X p the set of all bounded linear functionals on the space with asymmetric seminorm (X, p). As we did mention, X p is a convex cone in X # . One can define a norm | p on X p by (1.5) ϕ| p = sup{ϕ(x): x ∈ B p },ϕ ∈ X p . Some useful properties of this norm, whose proofs can be found in [9, 12], are collected in the following proposition. We agree to call a linear functional ϕ on (X, p), (p, ¯ p)-bounded if it is both p- and ¯ p-bounded, where ¯ p is the seminorm conjugate to p. Proposition 1.3. If ϕ is a bounded linear functional on a space with asym- metric seminorm (X, p),p =0, then the following assertions hold. (1) ϕ| p is the smallest of the numbers L ≥ 0 for which the inequality (1.4) holds. (2) We have ϕ| p = sup{ϕ(x)/p(x): x ∈ X, p(x) > 0} (1.6) = sup{ϕ(x): x ∈ X, p(x) < 1} (1.7) = sup{ϕ(x): x ∈ X, p(x)=1}. (1.8)
(3) If ϕ =0, then ϕ| p > 0. Also, if ϕ =0 and ϕ(x 0 )= ϕ| p for some x 0 ∈ B p , then p(x 0 )=1. (4) If ϕ is (p, ¯ p)-bounded, then ϕ(rB p )=(-rϕ| ¯ p ,rϕ| p ) and ϕ(rB ¯ p )=(-rϕ| p ,rϕ| ¯ p ) where B p = {x ∈ X : p(x) < 1},B ¯ p = {x ∈ X :¯ p(x) < 1} and r> 0. (5) If ϕ is p-bounded but not ¯ p-bounded, then ϕ(rB p )=(-∞,rϕ| p ). Remark 1.4. A linear functional ϕ : X → R is (p, ¯ p)-bounded if and only if (1.9) ∀x ∈ X, |ϕ(x)|≤ Lp(x), for some L ≥ 0. Indeed, if L 1 ,L 2 ≥ 0 are such that ϕ(x) ≤ L 1 p(x) and ϕ(x) ≤ L 2 p(-x), for all x ∈ X, then -ϕ(x)= ϕ(-x) ≤ L 2 p(x),x ∈ X, so (1.9) holds with L = max{L 1 ,L 2 }. Denote by X ¯ p the dual cone to (X, ¯ p) and let X * be the conjugate of the seminormed space (X, p s ), where p s is the symmetric seminorm associated to p and ¯ p (see (1.1)). Since ϕ(x) ≤ Lp(x) ≤ Lp s (x),x ∈ X, implies |ϕ(x)|≤ Lp(x),x ∈ X, it follows that X p is contained in the dual X * of (X, p s ). Similarly, X ¯ p is contained in X * too. For x * ∈ X * put x * = sup{x * (x): x ∈ X, p s (x) ≤ 1}. Then  is a norm on X * and X * is complete with respect to this norm, i.e. is a Banach space (even if p s is not a norm, see [11]). Proposition 1.5. Let (X, p) be a space with asymmetric seminorm. (1) The cones X p and X ¯ p are contained in X * and ϕ| p = ϕ,ϕ ∈ X p and ψ| ¯ p = ψ,ψ ∈ X ¯ p . (2) We have ϕ| p = - ϕ| ¯ p , so that ϕ ∈ X p and ϕ| p ≤ r ⇐⇒ -ϕ ∈ X ¯ p and - ϕ| ¯ p ≤ r. The properties of the dual space X p were investigated in [21] where, among other things, the analog of the weak * topology of X was defined. This is denoted by w and has a neighborhood basis at a point ϕ ∈ X p , the family V x 1 ,...,xn; (ϕ)= {ψ ∈ X p : ψ(x k ) - ϕ(x k ) < , k =1, ..., n},
for n ∈ N,x 1 , ..., x n ∈ X and > 0. The w -convergence of a net (ϕ i : i ∈ I ) in X p to ϕ ∈ X p can be characterized in the following way ϕ i w -→ ϕ ⇐⇒ ∀x ∈ X, ϕ i (x) → ϕ(x) in (R,u). It was shown that w is the restriction of the topology w * = σ(X * ,X ) on X * to X p (see [21]). This study was continued in [9] where separation theorems for convex sets and a Krein-Milman type theorem were proved. In [10] asymmetric locally convex spaces were introduced and their basic properties were studied. Another direction of investigation is that of best approximation in spaces with asymmetric seminorm. Due to the asymmetry of the seminorm we have two distances. For a nonempty subset Y of a space with asymmetric seminorm (X, p) and x ∈ X put (1.10) d p (x, Y ) = inf {p(y - x): y ∈ Y }, and (1.11) d p (Y,x) = inf {p(x - y): y ∈ Y }. Note that d p (Y,x)= d ¯ p (x, Y ). Duality formulae and characterization results for best approximation in spaces with asymmetric norm were obtained in [5, 6, 9, 12, 34, 35]. The papers [32, 33, 39] are concerned with best approximation in spaces of semi- Lipschitz functions defined on asymmetric metric spaces (called quasi-metric spaces) in connection with the extension properties of these functions. In the papers [13, 24, 25, 36], supposing that p is the Minkowski functional p C of a bounded convex body C in a normed space (X, ), some generic existence results for best approximation with respect to the asymmetric norm p C were proved, extending similar results from the normed case. As in the symmetric case, the geometric properties of the body C (or, equivalently, of the func- tional p C ) are essential. A study of the moduli of convexity and smoothness corresponding to p C is done in [43]. Best approximation with respect to some asymmetric norms in concrete function spaces of continuous or of integrable functions, called sign-sensitive approximation, was also studied in a series of papers, see [14, 15, 16, 41], the references quoted therein, and the monograph by Krein and Nudelman [23, Ch. 9, §5]). The present paper, which can be viewed as a sequel to [12] and [9], is concerned mainly with characterizations of the elements of best approximation in a subspace Y of a space with asymmetric norm (X, p) and duality results for best approximation. As in the case of (symmetric) normed spaces the characterizations will be done in terms of some linear bounded functionals vanishing on Y . The duality results will involve the annihilator in X p of the subspace Y. For this reason we start by recalling some extension results for bounded linear functionals on spaces with asymmetric seminorm. For proofs, all resorting to the classical Hahn-Banach extension theorem, see [9, 12].
Theorem 1.6. Let (X, p) be a space with asymmetric seminorm and Y a linear subspace of X . If ϕ 0 : Y → R is a linear p-bounded functional on Y then there exists a p-bounded linear functional ϕ defined on the whole X such that ϕ| Y = ϕ 0 and ϕ| p = ϕ 0 | p . We agree to call a functional ϕ satisfying the conclusions of the above theorem a norm preserving extension of ϕ 0 . Based on this extension result one can prove the following existence result. Proposition 1.7. Let (X, p) be a space with asymmetric seminorm and x 0 ∈ X such that p(x 0 ) > 0. Then there exists a p-bounded linear functional ϕ : X → R such that ϕ| p =1 and ϕ(x 0 )= p(x 0 ). In its turn, this proposition has the following corollary. Corollary 1.8. If p(x 0 ) > 0 then p(x 0 ) = sup{ϕ(x 0 ): ϕ ∈ X p , ϕ| p ≤ 1}. Moreover, there exists ϕ 0 ∈ X p , ϕ 0 | p =1, such that ϕ 0 (x 0 )= p(x 0 ). The following proposition is the asymmetric analog of a well known result of Hahn. Proposition 1.9. ([12]) Let Y be a subspace of a space with asymmetric seminorm (X, p) and x 0 ∈ X . (1) If d := d p (x 0 ,Y ) > 0, then there exists ϕ ∈ X p such that (i) ϕ| Y =0, (ii) ϕ| p =1, and (iii) ϕ(-x 0 )= d. (2) If ¯ d := d p (Y,x 0 ) > 0, then there exists ψ ∈ X p such that (j) ψ| Y =0, (jj) ψ| p =1, and (jjj) ψ(x 0 )= ¯ d. 2. BEST APPROXIMATION IN SPACES WITH ASYMMETRIC SEMINORM Let (X, p) be a space with asymmetric seminorm, ¯ p the seminorm conjugate to p and Y a nonempty subset of X. The distances d p (x, Y ) and d p (Y,x) from an element x ∈ X to Y are defined by the formulae (1.10) and (1.11). An element y 0 ∈ Y such that p(y 0 - x)= d p (x, Y ) will be called a p-nearest point to x in Y , and an element y 1 ∈ Y such that p(x - y 1 )=¯ p(y 1 - x)= d ¯ p (x, Y ) is called a ¯ p-nearest point to x in Y. Denote by (2.1) P Y (x)= {y ∈ Y : p(y - x)= d p (x, Y )}, and ¯ P Y (x)= {y ∈ Y : p(x - y)= d p (Y,x)}, the possibly empty sets of p-nearest points, respectively ¯ p-nearest points, to x in Y . The set Y is called p-proximinal, p-semi-Chebyshev, p-Chebyshev if
for every x ∈ X the set P Y (x) is nonempty, contains at most one element, contains exactly one element, respectively. Similar definitions are given in the case of ¯ p-nearest points. A semi-Chebyshev set is called also a uniqueness set. For a nonempty subset Y of a space with asymmetric seminorm (X, p), denote by Y ⊥ p the annihilator of Y in X p , i.e. Y ⊥ p = {ϕ ∈ X p : ϕ| Y =0}. We start by a characterization of nearest points given in [12] we shall need in the sequel. Proposition 2.1. ([12]) Let (X, p) be a space with asymmetric seminorm, Y a subspace of X and x 0 a point in X . (1) Suppose that d := d p (x 0 ,Y ) > 0. An element y 0 ∈ Y is a p-nearest point to x 0 in Y if and only if there exists a bounded linear functional ϕ : X → R such that (i) ϕ| Y =0, (ii) ϕ| p =1, (iii) ϕ(-x 0 )= p(y 0 - x 0 ). (2) Suppose that ¯ d := d p (Y,x 0 ) > 0. An element y 0 ∈ Y is a ¯ p-nearest point to x 0 in Y if and only if there exists a bounded linear functional ψ : X → R such that (j) ψ| Y =0, (jj) ψ| p =1, (jjj) ψ(x 0 )= p(x 0 - y 0 ). From this theorem one can obtain characterizations of sets of nearest points. Corollary 2.2. Let (X, p) be a space with asymmetric seminorm, Y a subspace of X, x ∈ X, and Z a nonempty subset of Y . (1) If d = d p (x 0 ,Y ) > 0 then Z ⊂ P Y (x) if and only if there exists a functional ϕ ∈ X p such that (i) ϕ| Y =0, (ii) ϕ| p =1, (iii) ∀y ∈ Z, ϕ(-x 0 )= p(y - x 0 ). (2) If ¯ d = d p (Y,x 0 ) > 0 then Z ⊂ ¯ P Y (x) if and only if there exists a functional ψ ∈ X p such that (j) ψ| Y =0, (jj) ψ| p =1, (jjj) ∀y ∈ Z, ψ(x 0 )= p(x 0 - y). In the next proposition we extend to the asymmetric case some charac- terization results for semi-Chebyshev subspaces (see [42, Chapter I, Theorem 3.2]). Theorem 2.3. Let Y be a subspace of a space with asymmetric norm (X, p) such that p(x) > 0 for every x =0. Then the following assertions are equiva- lent. (1) Y is a p-semi-Chebyshev subspace of X . (2) There are no ϕ ∈ Y ⊥ p and x 1 ,x 2 ∈ X with x 1 - x 2 ∈ Y \{0}, such that (i) ϕ| p =1 and (ii) ϕ(-x i )= p(-x i ),i =1, 2.
(3) There are no ψ ∈ Y ⊥ p ,x ∈ X, and y 0 ∈ Y \{0} such that (j) ψ| p =1 and (jj) ψ(-x)= p(-x)= p(y 0 - x). Proof. (1) ⇒ (2) Suppose that (2) does not hold. Let ϕ ∈ Y ⊥ p and x 1 ,x 2 ∈ X with x 1 - x 2 ∈ Y \{0}, such that the conditions (i) and (ii) of the assertion (2) are satisfied, and put y 0 = x 1 - x 2 . Then ϕ(-x 2 )= p(-x 2 ) ⇐⇒ ϕ(y 0 - x 1 )= p(y 0 - x 1 ), and ϕ(-x 1 )= p(-x 1 ) ⇐⇒ ϕ(0 - x 1 )= p(0 - x 1 ). By Proposition 2.1, it follows that 0 and y 0 are p-nearest points to x 1 in Y . (2) ⇒ (3) Suppose that (3) does not hold. Then there exist ψ ∈ Y ⊥ p ,x ∈ X, and y 0 ∈ Y \{0} such that the conditions (j) and (jj) of the assertion (3) are fulfilled. It follows that the conditions (i) and (ii) of the assertion (2) are satisfied by ϕ = ψ, x 1 = x and x 2 = y 0 - x, i.e. (2) does not hold. (3) ⇒ (1) Supposing that (1) does not hold, there exist z ∈ X \ Y and y 1 ,y 2 ∈ Y, y 1 = y 2 , such that p(y 1 - z )= p(y 2 - z )= d p (z,Y ). If d p (z,Y )=0, then y 1 = y 2 = z, a contradiction which shows that d p (z,Y ) > 0. If x := z - y 1 , then d p (z - y 1 ,Y ) = inf {p(y + y 1 - z ): y ∈ Y } = inf {p(y - z ): y ∈ Y } = d p (z,Y ) = p(y 1 - z ) = p(y 2 - z ). By Proposition 2.1, there exists ψ ∈ Y ⊥ p , ψ| p =1, such that ψ(y 1 - z )= p(y 1 - z )= p(y 2 - z ), or, denoting y 0 := y 2 - y 1 , this is equivalent to ψ(-x)= p(-x)= p(y 0 - x), showing that (3) does not hold. Remark 2.4. Obviously that a similar characterization result holds for ¯ p- semi-Chebyshev subspaces. Using Corollary 2.2, one can extend Theorem 2.3 to obtain characterizations of pseudo-Chebyshev subspaces, a notion introduced by Mohebi [28] in the case of normed spaces. Concerning other weaker notions of Chebyshev spaces – quasi-Chebyshev subspaces, weak-Chebyshev subspaces, as well as for their behaviour in concrete function spaces, see the papers [26, 27, 29, 31]. For a subset Z of a vector space X denote by aff(Z ) the affine hull of the set Z , i.e.
aff(Z )= {x ∈ X : ∃n ∈ N, ∃ z 1 , ..., z n ∈ Z, ∃ a 1 , ..., a n ∈ R,a 1 + ... + a n = 1 such that x = a 1 z 1 + ... + a n z n }. There exists a unique subspace Y of X such that aff(Z )= z + Y, for an arbitrary z ∈ Z . By definition, the affine dimension of the set Z is the dimension of this subspace Y of X . A subspace Y of a space with asymmetric norm (X, p) is called p-pseudo- Chebyshev if it is p-proximinal and the set P Y (x) has finite affine dimension for every x ∈ X . The following theorem extends a result proved by Mohebi [28] in normed spaces. Theorem 2.5. Let Y be a subspace of an asymmetric normed space (X, p) such that p(x) > 0 for every x =0. The following assertions are equivalent. (1) The subspace Y is p-pseudo-Chebyshev. (2) There do not exist ϕ ∈ Y ⊥ p , x 0 ∈ X, and infinitely many linearly independent elements x n ∈ X, n ∈ N, with x 0 - x n ∈ Y, n ∈ N, such that (i) ϕ| p =1 and (ii) ϕ(-x n )= p(-x n ),n =0, 1, ... . (3) There do not exist ψ ∈ Y ⊥ p ,x 0 ∈ X, and infinitely many linearly independent elements y n ∈ Y, n ∈ N, such that (j) ψ| p =1 and (jj) ψ(-x 0 )= p(-x 0 )= p(y n - x 0 ),n =1, 2, ... . Proof. (1) ⇒ (2) Suppose that (2) does not hold. Then there exist ϕ ∈ Y ⊥ p ,x 0 ∈ X, and infinitely many linearly independent elements x n ∈ X, with x 0 - x n ∈ Y, n ∈ N, satisfying the conditions (i) and (ii). The elements y n := x 0 - x n ,n ∈ N, all belong to Y , are linearly independent, and ϕ(y n - x 0 )= ϕ(-x n )= p(-x n )= p(y n - x 0 ), so that, by Corollary 2.2, they are all contained in P Y (x 0 ), showing that Y is not p-pseudo-Chebyshev. (2) ⇒ (3) Suppose again that (3) does not hold, and let ψ ∈ Y ⊥ p ,x 0 ∈ X, and the linearly independent elements {y n : n =1, 2, ...}⊂ Y fulfilling the conditions (j) and (jj). Then x n := x 0 - y n ,n =1, 2, ..., are linearly independent elements in X and ψ(-x n )= ψ(y n - x 0 )= p(-x n ),n =0, 1, 2, ..., showing that (2) does not hold. (3) ⇒ (1) Supposing that (1) does not hold, there exist an element z ∈ X and an infinite set {y n : n =1, 2, ...} of linearly independent elements contained in P Y (z ). By Corollary 2.2, there exists ϕ ∈ Y ⊥ p , ϕ| p =1, such that ϕ(y n - z )= d p (z,Y )= p(y n - z ),n =1, 2, ... .
Putting x := z - y 1 we have d p (x, Y ) = inf {p(y + y 1 - z ): y ∈ Y } = inf {p(y - z ): y ∈ Y } = = d p (z,Y )= p(y n - z )= p(y n - y 1 - x),n =2, 3, ..., showing that {y n - y 1 : n =2, 3, ...}⊂ P Y (x). By Corollary 2.2, there exists ψ ∈ Y ⊥ p with ψ| p = 1 such that ψ(y n - y y - x 0 )= p(y n - y 1 - x 0 ),n =2, 3, ..., showing that (3) does not hold. Phelps [37] emphasized for the first time some close connections existing between the approximation properties of the annihilator Y ⊥ of a subspace Y of a normed space X and the extension properties of the subspace Y .A presentation of various situations in which such a connection occurs is done in [8]. The case of spaces with asymmetric norms was considered in [34, 35]. Let (X, p) be a space with asymmetric seminorm and Y a subspace of X . For a p-bounded linear functional ϕ : Y → R denote by E p (ϕ)= {ψ ∈ X p : ψ| Y = ϕ, ψ| p = ϕ| p }, the set of all norm-preserving extensions of the functional ϕ. By the Hahn- Banach theorem (Theorem 1.6) the set E p (ϕ) is always nonempty. For ϕ ∈ X p consider the following minimization problem (2.2) γ (ϕ, Y ⊥ p ) := inf {ϕ + ψ| p : ψ ∈ Y ⊥ p }. A solution to this problem is an element ψ 0 ∈ Y ⊥ p such that ϕ + ψ 0 | p = γ (ϕ, Y ⊥ p ). Denote by Π Y ⊥ p (ϕ) the set of all these solutions. Theorem 2.6. If the linear functional ϕ : X → R is (p, ¯ p)-bounded then the minimization problem (2.2) has a solution and the following formulae hold (2.3) γ (ϕ, Y ⊥ p )= ϕ| Y | p and Π Y ⊥ p (ϕ)= E p (ϕ| Y ) - ϕ. Proof. Let ϕ ∈ X p ∩ X ¯ p and ψ ∈ Y ⊥ p . Then ϕ + ψ| p ≥(ϕ + ψ)| Y | p = ϕ| Y | p , implying γ (ϕ, Y ⊥ p ) ≥ϕ| Y | p . If Φ ∈ E p (ϕ| Y ) then, because ϕ is (p, ¯ p)-bounded, -ϕ ∈ X p (see Propo- sition 1.5, ψ := Φ - ϕ ∈ Y ⊥ p , and γ (ϕ, Y ⊥ p ) ≤ϕ + ψ| p = Φ| p . Therefore γ (ϕ, Y ⊥ p )= ϕ| Y | p and E p (ϕ| Y ) - ϕ ⊂ Π Y ⊥ p (ϕ). Conversely, if ψ ∈ Π Y ⊥ p (ϕ), then Φ := ϕ + ψ satisfies Φ| Y = ϕ| Y and Φ| p = ϕ + ψ| p = γ (ϕ, Y ⊥ p )= ϕ| Y | p , i.e. Φ ∈ E p (ϕ| Y ) and ϕ +Π Y ⊥ p (ϕ) ⊂ E p (ϕ| Y ) ⇐⇒ Π Y ⊥ p (ϕ) ⊂ E p (ϕ| Y ) - ϕ.
Denoting by (2.4) Y ⊥ = {ψ ∈ X * : ψ| Y =0}, the annihilator Y ⊥ of a subspace Y of X in the symmetric dual X * of the seminormed space (X, p s ) * , it follows that Y ⊥ is a subspace of X * . Consider on X * the asymmetric extended norm | * p : X * → [0; ∞] defined by ϕ| * p = sup ϕ(B p ). We have for any ϕ ∈ X * ϕ ∈ X p ⇐⇒ ϕ| * p < ∞, and ϕ| * p = ϕ| p for ϕ ∈ X p (see Proposition 1.5). For ϕ ∈ X p consider the distance from ϕ to Y ⊥ defined by d p (Y ⊥ ,ϕ) = inf {ϕ - ψ| * p : ψ ∈ Y ⊥ }. Because ϕ - 0| * p = ϕ| p < ∞ this distance is always finite. Put P Y ⊥ (ϕ)= {ψ ∈ Y ⊥ : ϕ - ψ| p = d p (Y ⊥ ,ϕ)}. Theorem 2.7. Every ϕ ∈ X p has a ¯ p-nearest point in Y ⊥ and the following formulae hold d p (Y ⊥ ,ϕ)= ϕ| Y | p and P Y ⊥ (ϕ)= ϕ - E p (ϕ| Y ). Proof. For ψ ∈ Y ⊥ we have ϕ - ψ| * p ≥(ϕ - ψ)| Y | * p = ϕ| Y | p , implying d p (Y ⊥ ,ϕ) ≥ϕ| Y | p . If Φ ∈ E p (ϕ| Y ), then ψ := ϕ - Φ ∈ Y ⊥ and d p (Y ⊥ ,ϕ) ≤ϕ - ψ| * p = Φ| p = ϕ| Y | p . Therefore d p (Y ⊥ ,ϕ)= ϕ| Y | p and ϕ - E p (ϕ| Y ) ⊂ P Y ⊥ (ϕ). If ψ ∈ P Y ⊥ (ϕ) and Φ := ϕ - ψ, then Φ| Y = ϕ| Y and Φ| p = ϕ - ψ| p = d p (Y ⊥ ,ϕ)= ϕ| Y | p , i.e. Φ ∈ E p (ϕ| Y ), showing that ϕ - P Y ⊥ (ϕ) ⊂ E p (ϕ| Y ), or equivalently, P Y ⊥ (ϕ) ⊂ ϕ - E p (ϕ| Y ). From these theorems we obtain some uniqueness conditions for the mini- mization problems we have considered, in terms of the uniqueness of norm- preserving extensions. Corollary 2.8. Let (X, p) be a space with asymmetric seminorm and Y a subspace of X . (1) If every f ∈ Y p has a unique norm preserving extension F ∈ X p , then the minimization problem (2.2) has a unique solution for every ϕ ∈ X p .
(2) Every point ϕ ∈ X p has a unique ¯ p-nearest point in Y ⊥ if and only if every f ∈ Y p has a unique norm-preserving extension F ∈ X p . Proof. (1) If every f ∈ Y p has a unique norm-preserving extension F ∈ X p , then for every ϕ ∈ X p the set Π Y ⊥ p (ϕ)= ϕ + E p (ϕ| Y ) contains exactly one element. (2) Similarly, P Y ⊥ (ϕ)= ϕ-E p (ϕ| Y ) contains exactly one element, provided every f ∈ Y p has exactly one norm-preserving extension F ∈ X p . Conversely, suppose that there exists f ∈ Y p having two distinct norm- preserving extensions F 1 ,F 2 ∈ X p . Then P Y ⊥ (F 1 )= F 1 - E p (F 1 | Y )= F 1 - E p (f ) ⊃{0,F 1 - F 2 }. Remark 2.9. We can not prove the reverse implication in the assertion (1) of the above corollary. To do this we would need an extension theorem for (p, ¯ p)-bounded linear functionals, preserving both p- and ¯ p-norm, and we are not aware of such a result. Some results connecting the -approximations and -extensions were ob- tained by Rezapour [38]. In the next proposition we transpose these results to the asymmetric case. Let (X, p) be a space with asymmetric seminorm and Y a subspace of X . For x ∈ X and > 0 let P Y (x)= {y ∈ Y : p(y - x) ≤ d p (x, Y )+ } and ¯ P Y (x)= {y ∈ Y : p(x - y) ≤ d p (Y,x)+ } denote the nonempty sets of -p-, respectively -¯ p-nearest points to x in Y . For ϕ ∈ X p consider the set of -solutions of the minimization problem (2.2) Π Y ⊥ p (ϕ)= {ψ ∈ Y ⊥ p : ϕ + ψ| p ≤ γ (ϕ, Y ⊥ p )+ } and, finally, denote by E p (f )= {F ∈ X p : F | Y = f and F | p ≤f | p + }, the set of -extensions of a functional f ∈ Y p . These two sets are related in the following way. Proposition 2.10. Let (X, p) be a space with asymmetric seminorm, Y a subspace of X and ϕ ∈ X p . Then Π Y ⊥ p (ϕ)= E p (ϕ| Y ) - ϕ.
Proof. Indeed, by Theorem 2.6, ψ ∈ Π Y ⊥ p (ϕ) ⇐⇒ ψ ∈ Y ⊥ p and ϕ + ψ| p ≤ γ (ϕ, Y ⊥ p )+ = ϕ| Y | p + ⇐⇒ ϕ + ψ ∈ E p (ϕ| Y ). Working with the annihilator Y ⊥ of the subspace Y in the symmetric dual X * =(X, p s ) * given by (2.4) and putting ¯ P Y ⊥ (ϕ)= {ψ ∈ Y ⊥ : ϕ - ψ| p ≤ d p (ϕ, Y ⊥ )+ }, we have Proposition 2.11. Let Y be a subspace of a space with asymmetric semi- norm (X, p),> 0, and ϕ ∈ X p . Then ¯ P Y (ϕ)= ϕ - E p (ϕ| Y ). Proof. Indeed, by Theorem 2.7, ψ ∈ ¯ P Y ⊥ (ϕ) ⇐⇒ ψ ∈ Y ⊥ and ϕ - ψ| p ≤ d p (Y ⊥ ,ϕ)+ = ϕ| Y | p + ⇐⇒ ϕ - ψ ∈ E p (ϕ| Y ). REFERENCES [1] Alegre, C. Ferrer, J. and Gregori, V., Quasi-uniformities on real vector spaces, Indian J. Pure Appl. Math., 28, no. 7, pp. 929–937, 1997. [2] , On the Hahn-Banach theorem in certain linear quasi-uniform structures, Acta Math. Hungar., 82, no. 4, pp. 325–330, 1999. [3] Alimov, A. R., The Banach-Mazur theorem for spaces with nonsymmetric distance, Uspekhi Mat. Nauk, 58, no. 2, pp. 159–160, 2003. [4] Babenko, V. F., Nonsymmetric approximations in spaces of summable functions, Ukrain. Mat. Zh., 34, no. 4, pp. 409–416, 538, 1982. [5] , Nonsymmetric extremal problems of approximation theory, Dokl. Akad. Nauk SSSR, 269, no. 3, pp. 521–524, 1983. [6] , Duality theorems for certain problems of the theory of approximation, Current problems in real and complex analysis, Akad. Nauk Ukrain. SSR Inst. Mat., Kiev, pp. 3–13, 148, 1984. [7] Borodin, P. A., The Banach-Mazur theorem for spaces with an asymmetric norm and its applications in convex analysis, Mat. Zametki, 69, no. 3, pp. 329–337, 2001. [8] Cobzas ¸, S., Phelps type duality results in best approximation, Rev. Anal. Num´er. Th´eor. Approx., 31, no. 1, pp. 29–43, 2002. [9] , Separation of convex sets and best approximation in spaces with asymmetric norm, Quaest. Math., 27, no. 3, 275–296, 2004. [10] , Asymmetric locally convex spaces, Int. J. Math. Math. Sci., no. 16, 2585–2608, 2005. [11] Cobzas ¸, S. and Must˘ at¸a, C., Extension of bilinear functionals and best approximation in 2-normed spaces, Studia Univ. Babe¸s-Bolyai, Mathematica, 43, pp. 1–13, 1998. [12] , Extension of bounded linear functionals and best approximation in spaces with asymmetric norm, Rev. Anal. Num´er. Th´eor. Approx., 33, no. 1, pp. 39–50, 2004.
[13] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx. Theory, 94, no. 1, pp. 54–72, 1998. [14] Dolzhenko, E. P. and Sevastyanov, E. A., Approximations with a sign-sensitive weight (existence and uniqueness theorems), Izv. Ross. Akad. Nauk Ser. Mat., 62, no. 6, pp. 59–102, 1998. [15] , Sign-sensitive approximations, J. Math. Sci. (New York), 91, no. 5, pp. 3205– 3257, 1998. [16] , Approximation with a sign-sensitive weight (stability, applications to snake the- ory and Hausdorff approximations), Izv. Ross. Akad. Nauk Ser. Mat., 63, no. 3, pp. 77–118, 1999. [17] Ferrer, J., Gregori, V. and Alegre, C., Quasi-uniform structures in linear lattices, Rocky Mountain J. Math., 23, no. 3, pp. 877–884, 1993. [18] Garc´ıa-Raffi, L. M., Romaguera, S. and S´ anchez P´ erez, E. A., Extensions of asymmetric norms to linear spaces, Rend. Istit. Mat. Univ. Trieste, 33, nos. 1–2, 113– 125, 2001. [19] , The bicompletion of an asymmetric normed linear space, Acta Math. Hungar., 97, no. 3, pp. 183–191, 2002. [20] , Sequence spaces and asymmetric norms in the theory of computational com- plexity, Math. Comput. Modelling, 36, nos. 1–2, pp. 1–11, 2002. [21] , The dual space of an asymmetric normed linear space, Quaest. Math., 26, no. 1, pp. 83–96, 2003. [22] , On Hausdorff asymmetric normed linear spaces, Houston J. Math., 29, no. 3, pp. 717–728 (electronic), 2003. [23] Krein, M. G. and Nudelman, A. A., The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973 (in Russian), English translation: American Mathe- matical Society, Providence, R.I., 1977. [24] Chong Li, On well posed generalized best approximation problems, J. Approx. Theory, 107, no. 1, pp. 96–108, 2000. [25] Chong Li and Renxing Ni, Derivatives of generalized distance functions and existence of generalized nearest points, J. Approx. Theory, 115, no. 1, pp. 44–55, 2002. [26] Mohebi, H., On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory, 107, no. 1, pp. 87–95, 2000. [27] , Pseudo-Chebyshev subspaces in L 1 , Korean J. Comput. Appl. Math., 7, no. 2, pp. 465–475, 2000. [28] , On pseudo-Chebyshev subspaces in normed linear spaces, Math. Sci. Res. Hot- Line, 5, no. 9, pp. 29–45, 2001. [29] , Quasi-Chebyshev subspaces in dual spaces, J. Nat. Geom., 20, nos. 1-2, pp. 33–44, 2001. [30] , On pseudo-Chebyshev subspaces in normed linear spaces, J. Nat. Geom., 24, nos. 1–2, pp. 37–56, 2003. [31] Mohebi, H. and Rezapour, Sh., On weak compactness of the set of extensions of a continuous linear functional, J. Nat. Geom., 22, nos. 1–2, pp. 91–102, 2002. [32] Must˘ at¸a, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numer. Theor. Approx., 30, no. 1, pp. 61–67, 2001. [33] , On the extremal semi-Lipschitz functions, Rev. Anal. Num´er. Th´eor. Approx., 31, no. 1, pp. 103–108, 2002. [34] , A Phelps type theorem for spaces with asymmetric norms, Bul. S¸tiint ¸. Univ. Baia Mare, Ser. B, Matematic˘ a-Informatic˘ a, 18, no. 2, pp. 275–280, 2002. [35] , On the uniqueness of the extension and unique best approximation in the dual of an asymmetric linear space, Rev. Anal. Numer. Theor. Approx., 32, no. 2, pp. 187–192, 2003.
[36] Renxing Ni, Existence of generalized nearest points, Taiwanese J. Math., 7, no. 1, pp. 115–128, 2003. [37] Phelps, R. R., Uniqueness of Hahn-Banach extensions and best approximations, Trans. Amer. Marth Soc., 95, pp. 238–255, 1960. [38] Rezapour, Sh., -weakly Chebyshev subspaces of Banach spaces, Anal. Theory Appl., 19, no. 2, pp. 130–135, 2003. [39] Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, no. 2, pp. 292–301, 2000. [40] Romaguera, S. and Schellekens, M., Duality and quasi-normability for complexity spaces, Appl. Gen. Topol., 3, no. 1, pp. 91–112, 2002. [41] Simonov, B. V., On the element of best approximation in spaces with nonsymmetric quasinorm, Mat. Zametki, 74, no. 6, pp. 902–912, 2003. [42] Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Sub- spaces, Editura Academiei Romˆ ane and Springer-Verlag, Bucharest-New York-Berlin, 1970. [43] Zanco, C. and Zucchi, A., Moduli of rotundity and smoothness for convex bodies, Bolletino U. M. I., (7), 7-B, pp. 833–855, 1993. Received by the editors: March 9, 2006.
2006

Related Posts