BEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction

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1 Bull Korean Math Soc 43 (2006), No 2, pp BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of representation for continuous linear functionals on -inner product spaces in terms of best approximations and orthogonalities are given 1 Introduction Due to its applications, eg, in optimization, the problem of best approximation has a long history and gives rise to a lot of notions and techniques useful in functional analysis The usual framework of this theory consists in Banach or Hilbert spaces because the geometry of these spaces, via Birkhoff orthogonality or orthogonality with respect to the inner product, yields the support for results of existence and uniqueness for elements of best approximation Also, the study of the dual of these spaces offers remarkable information For a huge part of this theory the monograph of I Singer [11] is a good reference In manifold setting, namely Riemannian spaces as the natural generalization of Hilbert spaces, the problem of best approximation is treated in M Crâşmăreanu [1] S S Dragomir [4], S S Kim, Y J Cho and T D Narang [10] have given some characterizations of best approximation theorems for continuous linear functionals in terms of norm derivatives for classical framework Recently, M Crâşmareanu and S S Dragomir [2, 3] introduced the notion of -inner product as a generalization of classical inner products Received March 16, Mathematics Subject Classification: 26D15, 26D99, 46B15, 46B20 Key words and phrases: -inner product, -normed space, -orthogonality, best approximation, proximinal set *This work was supported by Korea Research Foundation Grant (KRF C00015)

2 378 Seong Sik Kim and Mircea Crâşmăreanu and obtained some properties eg, uniformly convexity and Gâteaux differentiability In this paper, some characterizations of representation for continuous linear functionals on -inner product spaces in terms of best approximation and orthogonalities are given 2 Preliminaries Definition 21 [3] Let X be a real linear space and k be a natural number A mapping (,, ) : X = X X R satisfies the }{{} times following conditions: (i) (α 1 x 1 +α 2 x 2, x 3,, x +1 ) = α 1 (x 1, x 3,, x +1 )+α 2 (x 2, x 3,, x +1 ) for all α 1, α 2 R, (ii) (x σ(1),, x σ() ) = (x 1,, x ) for all permutation σ of the indices (1,, ), (iii) (x,, x) > 0 if x 0, (iv) Cauchy-Buniakowski-Schwarz s inequality (in short, CBS inequality) (x 1,, x ) i=1 (x i,, x i ) with equality if and only if x 1,, x are linearly dependent Then (,, ) is called a -inner product and the pair (X, (,, )) a -inner product space For k = 1 we have the usual notion of an inner product, and for k = 2 we obtain the concept of Q-inner product from [5]-[7] Also, it follows that (0, x 2,, x ) = 0 and (αx 1,, αx ) = α (x 1,, x ) Examples (1) Let X = R n and x j = (x 1 j,, xn j ) Rn Then (x 1,, x ) = n ( x i j i=1 j=1 (2) Let (Ω, A, µ) be a measure space consisting of a set Ω, a σ-algebra A of subsets of Ω, and a countably additive and positive measure µ on A with µ(ω) < Then on X = L (Ω, A, µ) and x j X, 1 j, )

3 we have the -inner product Best approximations and orthogonalities 379 (x 1,, x ) = Ω i=1 x i (t)dµ(t) Theorem 22 If (X, (,, )) is a -inner product space and a mapping : X R is defined by x = (x,, x) 1/, then is a norm on X Proof Using the property (iv) of Definition 11, x + y = i=0 i=0 which give the triangle inequality ( )( x,, x, i } {{ } i times ( ) x i i y i = ( x + y ), y,, y }{{} i times x + y x + y for x, y X ) On the other hand, x 0 for all x X and x x = 0 Finally, we also have = 0 implies αx = α x, where α R and x X Thus, x is a norm on X This completes the proof Corollary 23 If (X, (,, )) is a -inner product space with the norm x = (x,, x) 1/, then the norm x satisfies the following generalization of parallelogram identity: x + y + x y = 2 k i=0 ( )( x,, x, 2(k i) }{{} 2i times y,, y }{{} 2(k i) times )

4 380 Seong Sik Kim and Mircea Crâşmăreanu Definition 24 [2] A real normed linear space is said to be a normed space if its norms is defined by a -inner product Recall that a normed linear space (X, ) is said to be uniformly convex if given ɛ > 0 there is a δ > 0 such that x 1, y 1, and x + y > 2 δ, then x y < ɛ Theorem 25 convex If X is a -normed space, then X is uniformly Proof Let ɛ > 0 be given Then there is a δ > 0 such that 2 (2 δ) < ɛ Let x 1 and y 1 and x + y > 2 δ Then, by the parallelogram identity and the CBS inequality, we have k ( )( ) x + y + x y = 2 x,, x, y,, y 2(k i) }{{}}{{} i=0 2i times 2 2(k i) times and so x y 2 x+y < ɛ Therefore we have x y < ɛ This completes the proof Recall that on a normed space (X, ) an element x X is said to be Birkhoff orthogonal to an element y X (write x B y) if x+λy x for all λ R ([8]) Definition 26 Let (X, (,, )) be a -inner product space and x, y X An element x X is said to be -orthogonal to an element y X (write x y) if (x,, x, y) = 0 Definition 27 Let X be a -normed space and ɛ [0, 1) An element x X is said to be ɛ-cbs-orthogonal to the element y X (write x CBS y(ɛ)) if (y, x,, x) ɛ x 1 y for all x, y X If ɛ = 0, then x is -orthogonal to y Let X be a -inner product space Fixed an element y in X and consider the functional f : X R defined by f(x) = (x, y,, y) for all x X Then f is a continuous linear functional on X and f(x) y 1 x for all x X Also, we have f y 1 On the other hand, we have f y f(y) = y Therefore, we have f = y 1

5 Best approximations and orthogonalities Orthogonalities and best approximations Theorem 31 The norm of a -normed space X is a Gâteaux differentiable with ( ) τ(x, y) = lim t 0 x + ty x t = (x,, x, y) x 1 for x 0 Proof Let x, y X with x 0 and t is a nonzero real number Since we have 1 ( x + ty + x t ) = 1 t Also, since we have = 1 t lim t 0 1 i=0 x + ty x t 1 t ( x + ty x ) ( )( i x + ty x ( x + ty k + x k ) k i=1 x + ty x lim t 0 t This completes the proof = ) x,, x, ty,, ty, }{{}}{{} i times i times = (x,, x, y) x + ty k i (x,, x, y) 2 x k k x k 1 =, x i 1 (x,, x, y) x 1 Lemma 32 If (X, (,, )) is a -inner product space, then the -orthogonality is equivalent with the Birkhoff orthogonality Proof If x y and x 0, then since τ(x, y) = 0, x B y Conversely, if x B y, then since τ(x, y) = 0 and a -inner product space is smooth, x y This completes the proof Let X be a normed linear space and G a nondense subspace of X If x X \ G and g o G, then an element g o is said to be the best approximation element of x in G if x g o x g

6 382 Seong Sik Kim and Mircea Crâşmăreanu for all g G The set of all elements of best approximation of x in G is denoted by P G (x) A proper linear subspace E of X is called proximinal in X if, for every x X, the set P G (x) contains at least one element The following theorems is well-known ([11]): Theorem 33 Let (X, ) be a normed linear space, G a linear subspace of X, x o X \ G and g o G Then g o P G (x o ) if and only if (x g o ) B G Theorem 34 Let X be a normed linear space and H be a hyperplane in X such that 0 H Then H is proximinal in X if and only if there exists y X \ {0} such that y B H In [8], it is know that x B y on a normed linear space if and only if there exists a continuous linear functional f such that f = 1, f(x) = x and f(y) = 0 Theorem 35 Let X be -normed space, G a linear subspace in X and x X \ G If g o P G (x), then there exists a continuous linear functional f : X R such that (1) f(x) = x g o (2) f(g) = 0 for all g G (3) f = 1 Proof Suppose that g o P G (x) Then x g o x g for all g G and there exists a continuous linear functional f o on X such that f o = 1 x g o, f o (g) = 0 for all g G and f o (x) = 1 Put f = x g o f o Then f satisfies the conditions (1), (2) and (3) This completes the proof Corollary 36 Let X be a -normed product, G a linear subspace in X and x X \ G The following statement are equivalent: (1) g o P G (x), (2) τ(x g o, g) = 0 for all g G, where τ is given by ( ), (3) There exists a unique continuous linear functional f : X R with the properties: f(x g o ) = x g o, f(g) = 0, f(y) y for all y X

7 Best approximations and orthogonalities 383 Theorem 37 Let (X, (,, )) be a complete -inner product space and f be a nonzero continuous linear functional on X, g o Ker(f) and x o X \ Ker(f) Then the following statements are equivalent: (1) g o P Ker(f) (x o ) (2) f(x) = f(x o) (x, x o g o,, x o g o ) for all x X x o g o Proof (1) (2): If g o P Ker(f) (x o ), then (x o g o ) Ker(f) Let w o = x o g o Then, since f(x)w o f(w o )x Ker(f) for all x X we have w o (f(x)w o f(w o )x), which implies for all x X and so we have (2) f(x) = f(w o) w o (x, w o,, w o ) (2) (1): Suppose that (2) holds Then f f(x o) x o g o and so x o g o f(x o g) f x o g for all g Ker(f) Therefore g o P Ker(f) (x o ), that is, g o P Ker(f) (x o ) This completes the proof Corollary 38 Let (X, (,, )) be a complete -inner product space and f be a nonzero continuous linear functional on X Then the following statements are equivalent: (1) Ker(f) is proximinal in X (2) There exists at least one y f X \ {0} such that f(x) = (x, y f,, y f ) for all x X (3) There exists at least one v f X \ {0} such that f(v f ) = f v f 1 4 Representation of a continuous linear functional Let (X, (,, )) be a -inner product space If Y is a non-empty subset of X, then the set Y CBSɛ = {y X : y CBS x(ɛ) for all x Y }

8 384 Seong Sik Kim and Mircea Crâşmăreanu is called the ɛ CBS-orthogonal complement of Y The following results is given by [9]: Lemma 41 Let X be a -inner product space, Y a non-empty closed proper subspace of X and ɛ [0, 1) Then Y CBSɛ is non-empty Theorem 42 Let X be a complete -inner product space, Y a non-empty closed subspace of X and ɛ [0, 1) Then we have X = Y + Y CBSɛ for all ɛ [0, 1) Theorem 43 Let (X, (,, )) be a complete -inner product space and f be a continuous linear functional on X Then there exists y f X such that f(x) = (x, y f,, y f ) for all x X and f = y f 1 Proof If f = 0, then y = 0 If f 0, then we let x o X with f(x o ) 0 Since the subspace Y = Ker(f) is closed in X, there is a unique y o Ker(f) and a unique z o Ker(f) such that x o = y o +z o It results that z o / Ker(f) Let λ R with λ 1 = f(x o) z o and y = λz o Since f(x)z o f(z o )x Ker(f) for all x X we have z o (f(x)z o f(z o )x), that is, which implies (f(x)z o f(z o )x, z o,, z o ) = 0, f(x) = f(z o) z o (x, z o,, z o ) = λ 1 (x, z o,, z o ) = (x, λz o,, λz o ) = (x, y,, y) for all x X Now, using CBS inequality, f(x) x y 1 y 1 And letting x = y y, we have f(x) = ( y y, y,, y) = 1 y (y, y,, y) = y 1 and so f

9 Best approximations and orthogonalities 385 Thus, we have f = y f 1 This completes the proof Corollary 44 Let (X, (,, )) be a complete -inner product space and f be a continuous linear functional on X such that S Ker(f) = {g Ker(f) : g 1} is weakly sequentially compact in X Then there exists y f X such that f(x) = (x, y f,, y f ) for all x X and f = y f 1 Theorem 45 Let X be a complete -inner product space, Y a non-empty closed subspace of X and X Y Then for all ɛ > 0, there exists a continuous linear functional f ɛ such that f ɛ 1 and f ɛ Y ɛ where f ɛ Y = sup{ f ɛ (x) : x = 1 for all x Y } Proof Let ɛ 1 Then the statement is trivial Assume that ɛ (0, 1) By Lemma 41, there exists a nonzero element y ɛ Y CBSɛ such that (x, y ɛ,, y ɛ ) ɛ x y ɛ 1 for all x X Taking x ɛ = y ɛ / y ɛ, we have x ɛ 1 The functional f ɛ : X R is defined by f ɛ (x) = (x, x ɛ,, x ɛ ) = 1 y 1 (x, y ɛ,, y ɛ ) Then we have f ɛ 1 and f ɛ Y ɛ This completes the proof Theorem 47 Let X be a complete -inner product space, f be a non-zero continuous linear functional on X Then for all ɛ > 0, there exists a non-zero element x ɛ X such that for all x X f(x) (x, x ɛ,, x ɛ ) ɛ x Proof Since f is a non-zero continuous linear functional on X, the subspace Y = Ker(f) is closed in X and Y X Let ɛ > 0 and δ(ɛ) = ɛ/(2 f ) If δ(ɛ) 1, then by Lemma 41, there exists en element y ɛ Y CBS(ɛ) such that ( ) (x, y ɛ,, y ɛ ) δ(ɛ) y y ɛ 1

10 386 Seong Sik Kim and Mircea Crâşmăreanu for all x X Let 0 < δ(ɛ) < 1 Then there exists an element y ɛ Y CBSɛ such that ( ) holds Putting w ɛ = y ɛ / y ɛ, we have f(x)w ɛ f(w ɛ )x Y and by ( ) (f(x)w ɛ f(w ɛ )x, w ɛ,, w ɛ ) δ(ɛ) f(x)w ɛ f(w ɛ )x w ɛ 1 2δ(ɛ) f x = ɛ x for all x X On the other hand, we have Putting x 1 ɛ = f(y ɛ) (f(x)w ɛ f(w ɛ )x, w ɛ,, w ɛ ) = f(x) w ɛ f(w ɛ )(x, w ɛ,, w ɛ ) = f(x) f(y ɛ) y ɛ (x, y ɛ,, y ɛ ) y ɛ y ɛ, we have (f(x)w ɛ f(w ɛ )x, w ɛ,, w ɛ ) = f(x) (x, x ɛ,, x ɛ ) and This completes the proof f(x) (x, x ɛ,, x ɛ ) ɛ x References [1] M Crâşmăreanu, Projections on complete, finite-dimensional Riemannian manifolds, An Univ Timişoara, Ser Mat-Inform 33 (1995), [2] M Crâşmăreanu and S S Dragomir, -inner products and -Riemannian Metrics, RGMIA Res Rep Coll 18 (2000) [3], -inner products in real linear spaces, Demonstratio Math 35 (2002), no 3, [4] S S Dragomir, A characterization of elements of best approximation in real normed linear spaces, Studia Univ Babeş-Bolyai Math 33 (1988), no 3, [5], Repersentation of continuous linear functionals on complete SQ-innerproduct spaces, An Univ Timişoara Ser Ştiinţ Mat 30 (1992), [6] S S Dragomir and I Muntean, Linear and continuous functional on complete Q-inner product spaces, Babeş-Bolyai Univ Sem Math Anal 7 (1987), [7] S S Dragomir and N M Ionescu, New properties of Q-inner-product spaces, Zb Rad (Kragujevac) 14 (1993), [8] R C James, Orthogonality and linear functionals in normed linear spaces, Trans Amer Math Soc 61 (1947),

11 Best approximations and orthogonalities 387 [9] S S Kim, Y J Cho, and S S Dragomir, Orthogonalities and decomposition theorems for -normed spaces and applications, preprint [10] S S Kim, Y J Cho, and T D Narang, Some remarks on orthogonality and best approximation, Demonstratio Math 30 (1997), no 2, [11] I Singer, Best Approximation in Normed Linear Spaces by elements of Linear Subspaces, Springer-Verlag, New York, 1970 Seong Sik Kim, Department of Mathematics, Dongeui University, Pusan , Korea sskim@deuackr Mircea Crâşmăreanu, Faculty of Mathematics, University AlICuza, Iaşi 6600, Romania mcrasm@uaicro