A Review on 2-normed Structures

Transcription

1 Int. Journal of Math. Analysis, Vol. 1, 2007, no. 4, A Review on 2-normed Structures Mehmet Açıkgöz University of Gaziantep, Science and Arts Faculty Department of Mathematics, Gaziantep, Turkey acikgoz@gantep.edu.tr Abstract. We will review the theory of 2-normed spaces and their structure and we will explain difference of this structure with the normed spaces one. Also, we will introduce a new structure called generalized 2-normed spaces. Mathematics Subject Classification: 46A15, 41A65 Keywords: 2-normed space, generalized 2-normed space, 2-subadditive, proximinal 1. Introduction We know that each construction is valuable for review. We want to review the structure of 2-normed spaces. The concept of 2-normed spaces has been investigated by Gahler in 1965 [2] and has been developed extensively in different subjects by others, for example [1, 2, 3, 4]. Let X be a vector space of dimension greater than one over Φ,where Φ is the real or complex numbers field. Suppose N (.,.) be a non-negative real function on XxX satisfies the following conditions: i) N (x, y) = 0 if and only if x and y are linearly dependent vectors ii) N (x, y) =N (y, x) for all x, y X iii) N (λx, y) = λ N (x, y) for all λ Φ and all x, y, z X iv) N (x + y, z) N (x, z)+n (y, z) for all x, y, z X Then N (.,.) is called a 2-norm on X and (X, N (.,.)) is called a linear 2- normed space. Every 2-normed space is a locally convex topological vector space [6]. In fact for a fixed b X, P b (x) =N (x, b), for all x X, is a seminorm and the family P = {P b } b X. First we give some examples here. Example 1. Let X = R 3 and consider the following 2-norm on X : N (x, y) = xxy = det i j k x 1 x 2 x 3 y 1 y 2 y 3,

2 188 Mehmet Açıkgöz where x =(x 1,x 2,x 3 ) and y =(y 1,y 2,y 3 ). Then (X, N (.,.)) is a 2-normed space. Example 2. Let P n denotes the set of all real polynomials of degree n, on the interval [0, 1]. By considering usual addition and scalar multiplication, P n is a linear vector space over the reals. Let {x 0,x 1,..., x 2n } be distinct fixed points in [0, 1] and define the following 2-norm on p n : N (f,g) = 2n k=0 f (x k ) g (x k ), whenever f and g are linearly independent. Then, (P n,n(.,.)) is a 2-normed space. Example 3. Let X = Q 3, the field be the rational number and consider the following 2-norm on X : N (x, y) = xxy = det i j k x 1 x 2 x 3 y 1 y 2 y 3, Then (X, N (.,.)) is a 2-normed space. Definition 1. i) A sequence {x n } n 1 in a 2-normed space (X, N (.,.)) is called a Cauchy sequence if there exist two linearly independent elements y and z in X such that {N (x n,y)} and N (x n,z) are real Cauchy sequences. ii) A sequence {x n } n 1 in a 2-normed space (X, N (.,.)) is called convergent if there exists x X such that {N (x n x, y)} n 1 thends to zero for all y X. iii) A 2-normed space (X, N (.,.)) is called 2-Banach space if every Cauchy sequence is convergent. The examples 1.1 and 1.2 are 2-Banach spaces while the example 1.3 does not.( For details see [1] ) Lemma 1. [9] i) Every 2-normed space of dimension 2 is a 2-Banach space, when the underlying field is complete. ii) If {x n } n 1 is a sequence in 2-normed space (X, N (.,.)) and lim n N (x n x, y) =0, then lim n N (x n,y)=n(x, y). Here, we mention 2-functionals and one of their applications in approximation theory. Definition 2. Let (X, N (.,.)) be a 2-normed space. A map F : XxX Φ is called 2-functional if it satisfies the following conditions, i) F (x + y, c + d) = F (x, c)+f (x, d)+f (y, c)+f (y, d) for all x, y, c, d X. ii) F (λx, λ y) = λλ F (x, y), for x, y X and all λ, λ Φ.

3 A Review on 2-normed Structures 189 For a 2-functional F : XxX Φ, we can define its norm as follows N (F )=Inf {M 0 N (F (x, y)) M.N (x, y), for all x, y X}. Theorem 2. [9] Let (X, N (.,.)) be a 2-normed space, W be a subspace of X and b X. If x 0 X and δ = Inf {N (x 0 w, b)} > 0 w W then there exists a bounded 2-functional F : Xx <b> Φ such that F wx<b> =0, F (x 0,b)=1, N (F )= 1 δ. By using Theorem 1.7, we can give the following result. Lemma 3. Let (X, N (.,.)) be a 2-normed space, W a subspace of X, b X and x X\W, where W denotes the closure of W in the seminormed space (X, P b ).Then an element w 0 W satisfies N (x w 0,b)=InfN (x w, b) w W if and only if there exists a bounded 2-functional F : Xx <b> Φ such that F wx<b> =0, N (F )=1 and F (x 0 w 0,b)=N(x 0 w 0,b). 2. Generalized 2-Normed Spaces Unfortunately, there are not any connection between normed spaces and 2- normed spaces. In 1999, in [4, 5, 7], Zofia Lewandowska introduced generalized 2-normed spaces and there are appropriate connection between normed spaces and generalized 2-normed spaces, and each 2-normed space is a generalized 2-normed space. Definition 3. Let X and Y be linear spaces. A function N (.,.) :XxY [0, ) is called a generalized 2-norm if it satisfies the following conditions: i) N (x, y) =N (y, x) for all x, y X ii)n (λx, y) = λ.n (x, y) for all x, y X, and λ Φ iii)n (x + y, z) N (x, z)+n (y, z), for all x, y X If X = Y then the generalized 2- normed space is denoted by (X, N (.,.)). For example, let A be a Banach algebra and N (x, y) =N (xy) for all x, y A. Then (A, N (.,.)) is a generalized 2-normed space. Also, let (X, N (.)) be a normed space. Then N (x, y) =N (x).n (y), for all x, y X, is a 2-norm on XxX. So (X, N (.,.)) is a generalized 2-normed space. These examples show us that theory of generalized 2-normed spaces includes theory of normed spaces and theory of Banach algebras.

4 190 Mehmet Açıkgöz Definition 4. i) Let (XxY,N (.,.)) be a generalized 2-normed space, W 1 be a subspace of X and W 2 a subspace of Y. Then W 1 xw 2 is called 2-Proximal if for every (x, y) XxY there exists (w 0,g 0 ) W 1 xw 2 such that N (x w 0,y g 0 )=Inf {N (x w, y g) :(w, g) W 1 xw 2 }. In this case, (w 0,g 0 ) is called 2-best approximation of (x, y) in W 1 xw 2 and the set of all 2-best approximations of (x, y) in W 1 xw 2 is denoted by PW 2 1 xw 2 (x, y). We could consider the special case of X = Y and W 1 = W 2 and x = y. ii) Let (XxY,N (.,.)) be a generalized 2-normed space and f be a real valued map on XxY. Then, f is called a 2-subadditive if f (x 1 + x 2,y) f (x 1,y)+f (x 2,y), f (x, y 1 + y 2 ) f (x, y 1 )+f (x, y 2 ) for all x 1,x 2,x X and all y 1,y 2,y Y. Also f is called bounded if there exists a positive real number M such that f (x, y) M.N (x, y) for all (x, y) XxY. Then the norm of f defined by N (f) =Inf {M >0 f (x, y) M.N (x, y), for all (x, y) XxY }. Theorem 4. Let (XxY,N (.,.)) be a generalized 2-normed space, W 1 be a subspace of X, W 2 be a subspace of Y and (x, y) XxY. Then M PW 2 1 xw 2 (x, y) if and only if there exists a 2-subadditive map f : XxY R such that f W1 x{y}= f {x}xw2 = f W1 xw 2 =0, N (f) 1, and f (x m 1,y m 2 )=N(x m 1,y m 2 ) for all m 1,m 2 M. Example 4. Let X = Y = R 2,W 1 = {(x 1,x 2 ) X : x 1 = x 2 }, W 2 = {(y 1,y 2 ) Y : y 1 = y 2 } and define N (.,.) :XxY R by N ((x 1,x 2 ), (y 1,y 2 )) = x 1 y 2 x 2 y 1 for all (x 1,x 2 ) X, (y 1,y 2 ) Y. Then W 1 xw 2 is 2-Proximinal subspace of XxY. Example 5. Let W 1 and W 2 be proximinal subspaces of (X, N (.) 1 ) and (Y,N (.) 2 ) respectively. Then N (x, y) =N (x) 1.N (y) 2 is generalized 2-norm on XxY and P W1 (x) xp W2 (y) PW 2 1 xw 2 (x, y), and this shows that 2-proximinality is a generalization of proximinality in a sense.

5 A Review on 2-normed Structures 191 References [1] White, A., 2-Banach spaces, Math Nachr.42(1969), [2] Gahler, S., Lineare 2-normierte Raume, Math. Nachr.28(1964)1-43. [3] Rezapour, Sh., Quasi-Chebyshew Subspaces in generalized 2-normed spaces, International Journal of pure and applied mathematical sciences, Vol.2 No.1(2005), pp [4] Lewandowska, Z., Generalized 2-normed spaces, Stuspskie Prace Matematyczno- Fizyczne 1(2001), [5] Lewandowska, Z., Linear operators on generalized 2-normed spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 42(90)(1999), no.4, [6] Rudin, W., Functional Analysis, McGraw-Hill, New York, [7] Lewandowska, Z., On 2-normed sets, Glasnik Mat. Ser.III 38(58) (2003), no.1, [8] Cho, Y., Lin, P., Kim, S.S., Misiak, A., Theory of 2-Inner Product Spaces, Nova Science Publishers, [9] Freese, R., Cho, Y., Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Received: October 10, 2006