Khayyam J. Math. 1 (2015), no. 2, 219{229 n-DUAL SPACES ASSOCIATED TO A NORMED SPACE YOSAFAT E. P. PANGALELA Communicated by M. Filali Abstract. For a real normed space X, we study the n-dual space of (X; k·k) and show that the space is a Banach space. Meanwhile, for a real normed space X of dimension d ≥ n which satisfies property (G), we discuss the n-dual space of (X; k·;:::; ·kG), where k·;:::; ·kG is the G¨ahler n-norm. We then investigate the relationship between the n-dual space of (X; k·k) and the n-dual space of (X; k·;:::; ·kG). We use this relationship to determine the n-dual space of (X; k·;:::; ·kG) and show that the space is also a Banach space. 1. Introduction In the 1960's, the notion of n-normed spaces was introduced by G¨ahler[2,3, 4,5] as a generalisation of normed spaces. For every real normed space X of dimension d ≥ n, G¨ahlershowed that X can be viewed as an n-normed space by using the G¨ahler n-norm, which is denoted by k·;:::; ·kG. This n-norm is defined by using the set of bounded linear functionals on X. Since then, many researchers have studied operators and functionals on n-normed space X (see [1,6,9, 10, 11, 13, 14, 15]). In [13], the author and Gunawan introduced the concept of n-dual spaces. For every real normed space X of dimension d ≥ n, there are two n-dual spaces associated to X. The first is the n-dual space of (X; k·k), and the other is the p n-dual space of (X; k·;:::; ·kG). In case X is the l space for some 1 ≤ p < 1, the author and Gunawan have investigated and given the relationship between both n-dual spaces [13]. Here we provide an analogues result on more general normed spaces. For a real normed space X, we investigate the n-dual space of (X; k·k) by using the (n − 1)-dual space of (X; k·k) (Theorem 3.2). We then focus on a real normed Date: Received: 16 August 2015, Accepted: 04 December 2015. 2010 Mathematics Subject Classification. Primary 46B20; Secondary 46C05, 46C15, 46B99, 46C99. Key words and phrases. n-dual spaces, n-normed spaces, bounded linear functionals. 219 220 Y.E.P. PANGALELA space X of dimension d ≥ n which satisfies property (G) and discuss the relation- ship between the n-dual space of (X; k·k) and the n-dual space of (X; k·;:::; ·kG) (Theorem 4.5). It is interesting to observe that both the n-dual space of (X; k·k) and the n-dual space of (X; k·;:::; ·kG) are Banach spaces (Theorem 3.3 and Theorem 4.7). 2. Preliminaries Let n be a nonnegative integer and X a real vector space of dimension d ≥ n. We call a real-valued function k·;:::; ·k on Xn an n-norm on X if for all 0 x1; : : : ; xn; x 2 X, we have (1) kx1; : : : ; xnk = 0 if and only if x1; : : : ; xn are linearly dependent; (2) kx1; : : : ; xnk is invariant under permutation; (3) kαx1; x2; : : : ; xnk = jαj kx1; x2; : : : ; xnk for all α 2 R; and 0 0 (4) kx1 + x ; : : : ; xnk ≤ kx1; : : : ; xnk + kx ; : : : ; xnk. We then call the pair (X; k·;:::; ·k) an n-normed space. An example of an n-normed space is the lp space, where 1 ≤ p < 1, equipped with 1 X X p 1 kx ; : : : ; x k := ··· jdet (x )j p 1 n p n! ijk j1 jn p for x1; : : : ; xn 2 l (see [7, Section 3]). Another interesting example of n-normed spaces is the G¨ahler n-norm which was introduced in [3,4,5]. Let X be a real normed space of dimension d ≥ n, (1) and X the dual space of X. G¨ahlershowed that the function k·;:::; ·kG which is given by kx1; : : : ; xnkG := sup det [fj (xi)]i;j (1) fi2X ;kfik≤1 1≤i≤n for all x1; : : : ; xn 2 X, is an n-norm on X. Hence every real normed space X can be viewed as an n-normed space (X; k·;:::; ·kG). Let X be a real normed space of dimension d ≥ n. Any real-valued function f on Xn is called an n-functional on X. An n-functional f is multilinear if it satisfies two following properties: (1) f (x + y ; : : : ; x + y ) = P f (z ; : : : ; z ) and 1 1 n n zi2fxi;yig;1≤i≤n 1 n (2) f (α1x1; : : : ; αnxn) = α1 ··· αn−1f (x1; : : : ; xn) for all x1; : : : ; xn; y1; : : : ; yn 2 X and α1; : : : ; αn 2 R. For multilinear n-functionals f; h on X, we define an n-functional f + h by (f + h)(x1; : : : ; xn) := f (x1; : : : ; xn) + h (x1; : : : ; xn) for x1; : : : ; xn 2 X. Then f + h is also multilinear. On the other hand, we say f = h if f (x1; : : : ; xn) = h (x1; : : : ; xn) for x1; : : : ; xn 2 X. We call an n-functional f bounded on a real normed space (X; k·k) (respectively, an n-normed space (X; k·;:::; ·k)) if there exists a constant K > 0 such that jf (x1; : : : ; xn)j ≤ K kx1k · · · kxnk (respectively, jf (x1; : : : ; xn)j ≤ K kx1; : : : ; xnk ) n-DUAL SPACES ASSOCIATED TO A NORMED SPACE 221 for all x1; : : : ; xn 2 X. Let Sn denote the group of permutations of (1; : : : ; n). Recall from [13] that every bounded multilinear n-functional f on (X; k·;:::; ·k) is antisymmetric in the sense that f (x1; : : : ; xn) = sgn (σ) f xσ(1); : : : ; xσ(n) for x1; : : : ; xn 2 X and σ 2 Sn. Here sgn (σ) = 1 if σ is an even permutation, and sgn (σ) = −1 if σ is an odd permutation. Note that if f is antisymmetric then for any linearly dependent x1; : : : ; xn 2 X, we have f (x1; : : : ; xn) = 0. Remark 2.1. In general, we do not have the antisymmetric property for bounded mul- tilinear n-functionals on (X; k·k). The space of bounded multilinear n-functionals on (X; k·k) is called the n-dual space (n) (0) of (X; k·k) and denoted by X . For n = 0, we define X as R. The function k·kn;1 on X(n) where jf (x1; : : : ; xn)j kfkn;1 := sup x1;:::;xn6=0 kx1k · · · kxnk for f 2 X(n), defines a norm on X(n) and then X(n) is a normed space. Meanwhile, the n-dual space of (X; k·;:::; ·k) is the space of all bounded multilinear n-functionals on (X; k·;:::; ·k). This space is also a normed space with the following norm jf (x1; : : : ; xn)j kfkn;n := sup . kx1;:::;xnk6=0 kx1; : : : ; xnk Now let X; Y be real normed spaces. We write B (X; Y ) to denote the set of bounded linear operators from X into Y . The function k·kop where ku (x)k kukop := sup x6=0 kxk for every u 2 B (X; Y ) , is a norm on B (X; Y ). For simplification, we write B (X; Y ) ∗ to denote the normed space B (X; Y ) equipped with k·kop. Otherwise, if k·k is a norm function on B (X; Y ), we write (B (X; Y ) ; k·k∗) to denote the normed space B (X; Y ) equipped by the norm k·k∗. 3. The n-dual space of (X; k·k) In this section, we first identify the bounded multilinear n-functionals on (X; k·k) (Proposition 3.1). We then identify the n-dual space of (X; k·k) by using the (n − 1)- dual space of (X; k·k) (Theorem 3.2). Finally we show that the n-dual space of (X; k·k) is a Banach space (Theorem 3.3). Proposition 3.1. Let X be a real normed space of dimension d ≥ n and f a bounded (n−1) multilinear n-functional on (X; k·k). Then there exists uf 2 B(X; X ) such that for x1; : : : ; xn−1; z 2 X, f (x1; : : : ; xn−1; z) = (uf (z)) (x1; : : : ; xn−1) . Furthermore, kfkn;1 = kuf kop. Proof. Take z 2 X and define an (n − 1)-functional fz on X with fz (x1; : : : ; xn−1) := f (x1; : : : ; xn−1; z) 222 Y.E.P. PANGALELA (n−1) for x1; : : : ; xn−1 2 X. We show fz 2 X . Note that for x1; : : : ; xn−1; y1; : : : ; yn−1 2 X and α1; : : : ; αn−1 2 R, we have fz (x1 + y1; : : : ; xn−1 + yn−1) = f (x1 + y1; : : : ; xn−1 + yn−1; z) X = f (z1; : : : ; zn−1; z) zi2fxi;yig;1≤i≤n−1 X = fz (z1; : : : ; zn−1), zi2fxi;yig;1≤i≤n−1 fz (α1x1; : : : ; αn−1xn−1) = f (α1x1; : : : ; αn−1xn−1; z) = α1 ··· αn−1f (x1; : : : ; xn−1; z) = α1 ··· αn−1fz (x1; : : : ; xn−1), and jfz (x1; : : : ; xn−1)j = jf (x1; : : : ; xn−1; z)j ≤ kfkn;1 kzk (kx1k · · · kxn−1k) n−1 since f is bounded on (X; k·k). Hence fz : X ! R is multilinear and bounded; and (n−1) then fz 2 X . (n−1) Now define uf : X ! X with uf (z) := fz for z 2 X. We have to show (n−1) uf 2 B(X; X ). First we show that uf is linear. Take z1; z2 2 X and α; β 2 R. For every x1; : : : ; xn−1 2 X, we have (uf (αz1 + βz2)) (x1; : : : ; xn−1) = fαz1+βz2 (x1; : : : ; xn−1) = f (x1; : : : ; xn−1; αz1 + βz2) = f (x1; : : : ; xn−1; αz1) + f (x1; : : : ; xn−1; βz2) = αf (x1; : : : ; xn−1; z1) + βf (x1; : : : ; xn−1; z2) = αfz1 (x1; : : : ; xn−1) + βfz2 (x1; : : : ; xn−1) = (αuf (z1)) (x1; : : : ; xn−1) + (βuf (z2)) (x1; : : : ; xn−1) = (αuf (z1) + βuf (z2)) (x1; : : : ; xn−1) and uf (αz1 + βz2) = αuf (z1) + βuf (z2).