ON n-norms AND BOUNDED n-linear FUNCTIONALS IN A HILBERT SPACE

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1 ON n-norms AND BOUNDED n-linear FUNCTIONALS IN A HILBERT SPACE SM GOZALI 1, H GUNAWAN 2, O NESWAN 2 Abstract In this paper we discuss the concept of n-normed spaces In particular, we show the equality of four different formulas of n-norms in a Hilbert space In addition, we study the notion of bounded n-linear functionals on an n-normed space and present some results on it 1 Introduction Let X be a real vector space with dim(x) 2 A real-valued function, : X X R is called a 2-norm on X if the following conditions hold: A1 x, y = 0 if and only if x, y are linearly dependent; A2 x, y = y, x for all x, y X; A3 αx, y = α x, y for all α R, x, y X; A4 x, y + z x, y + x, z for all x, y, z X The pair (X,, ) is then called a 2-normed space The notion of 2-normed spaces has been extended to that of n-normed spaces So, from now on, let n be a nonnegative integer and X be a real vector space of dimension d n A real-valued function,, on X n = X X (n factors) satisfying the following four properties: B1 x 1,, x n = 0 if and only if x 1,, x n are linearly dependent; B2 x 1,, x n is invariant under permutation; B3 αx 1,, x n = α x 1,, x n, for any α R; B4 x 0 + x 1, x 2,, x n x 0, x 2,, x n + x 1, x 2,, x n, is called an n-norm on X, and the pair (X,,, ) is called an n-normed space If X is a normed space with dual X, then as formulated by Gähler [2] we may define an n-norm on X by f 1 (x 1 ) f n (x 1 ) x 1,, x n G := sup f j X, f j 1 f 1 (x n ) f n (x n ) = sup det [f j (x i )] f j X, f j 1 Meanwhile, if X is equipped with an inner product,, we can define the standard n-norm on X by x 1,, x n S := det [ x i, x j ] 2000 Mathematics Subject Classification 46B20, 46C05, 46C15, 46B99, 46C99 Key words and phrases n-normed spaces, bounded n-linear functionals 1

2 The value of x 1,, x n S is nothing but the volume of the n-dimensional parallelepiped spanned by x 1,, x n The concept of 2-normed spaces and, more generally, that of n-normed spaces, were initially developed by Gähler [1, 2, 3, 4] in the 1960 s Since then, many researchers have developed and obtained various results, see eg [6, 7, 9] for some recent results Related works can be found, for examples, in [8, 11] Our interest here is to study various formulas of n-norms, especially in a Hilbert space So far, we already have two formulas of n-norms, Gähler s formula and the standard one In the next section, we show that the two formulas are identical As it turns out, we also have two more definitions of n-norms, which have different formulas but are actually the same with the previous two In the last section, we study the notion of bounded n-linear functionals on n-normed spaces and present some facts, including that in the standard case 2 n-norms in a Hilbert space Hereafter, let X be a Hilbert space, unless otherwise stated By Riesz Representation Theorem, X = X and Gähler s formula on X reduces to x 1,, x n G = sup det [ x i, y j ] y j X, y j 1 Applying the generalized Cauchy-Schwarz inequality [10], we have x 1,, x n G Now, by Hadamard s inequality [5], and so we get sup x 1,, x n S y 1,, y n S y j X, y j 1 y 1,, y n S y 1 y n, x 1,, x n G x 1,, x n S Conversely, assuming that x 1,, x n are linearly independent, let x 1,, x n be the vectors obtained from x 1,, x n through the Gram-Schmidt orthogonalization process Then x 1,, x n S = x 1 x n Now, if y j = 1 x j, j = 1,, n, then by the properties of determinants, we have j x x 1, y 1 x 1, y n x x n, y 1 x n, y n = 1 1, x 1 x 1, x n x 1 x n x n, x 1 x n, x = x 1 x n n Hence we obtain x 1,, x n G x 1,, x n S Therefore, we conclude that Gähler s formula is identical with the standard one 2

3 Suppose that X is separable, and {e 1, e 2, } is a complete orthonormal set in X Then, any member x of X may be identified by the sequence ( x, e j ), which is in l 2 As in [6], we can define an n-norm on X by [ ] 1/2 1 x 1,, x n 2 := det [α ijk ] 2, j 1 j n where α ij := x i, e j By Parseval s identity, properties of determinants, and limiting arguments [6], this n-norm may be derived from the standard one Therefore, we have: Fact 1 On a separable Hilbert space, the three formulas of n-norms, namely,, G,,, S and,, 2 coincide In addition to the above formulas, we shall introduce another formula of n-norm which we state in the following proposition Proposition 2 The following function x 1,, x n D := defines an n-norm on X sup y 1,, y n X y 1,, y n S 1 x 1, y 1 x 1, y n Proof It is obvious that, if x 1,, x n are linearly dependent vectors, then we have x 1,, x n D = 0 Conversely, if x 1,, x n D = 0, then the rows of the matrix [ x i, y j ] are linearly dependent for all y 1,, y n X with y 1,, y n S 1 This happens only if x 1,, x n are linearly dependent Next, by the properties of determinants, we have the invariance of x 1,, x n D under permutation Furthermore, we have αx 1,, x n D = α x 1,, x n D for any α R Finally, for arbitrary elements x 0, x 1,, x n in X, we have x 0 + x 1, y 1 x 0 + x 1, y n x 0, y 1 x 0, y n = x 1, y 1 x 1, y n + Taking the supremums of both sides, we obtain This completes the proof x 0 + x 1,, x n D x 0,, x n D + x 1,, x n D Regarding the last formula, we have the following proposition Proposition 3 The two formulas,, G and,, D are identical 3

4 Proof Let y 1,, y n be elements in X Since y 1,, y n S y 1 y n, we have y 1,, y n S 1 whenever y j 1 for j = 1,, n It thus follows that Conversely, if 0 < y 1,, y n S inequality we have x 1,, x n G x 1,, x n D 1, then by the generalized Cauchy-Schwarz det [ x i, y j ] det [ x i, x j ] det [ y i, y j ] = x 1,, x n S y 1,, y n S x 1,, x n S = x 1,, x n G Hence we obtain x 1,, x n D x 1,, x n G Therefore the two formulas are identical Corollary 4 On a separable Hilbert space, the four formulas of n-norms,,, G,,, S,,, 2, and,, D, are identical The last formula makes us realize that we can actually define an n-norm on the dual of an n-normed space Recall that we can define a norm on the dual X of a normed space (X, ) by the formula f := sup f(x), f X x 1 Now, if X is equipped with an n-norm,, and X denotes the dual of X, we would like to induce an n-norm on X from the n-norm,, on X It turns out that it is possible to do so, as it is shown in the following proposition The proof is similar to that of Proposition 2, and so we do not repeat it here Proposition 5 Let (X,,, ) be an n-normed space The function,, : (X ) n R given by the formula f 1 (x 1 ) f n (x 1 ) f 1,, f n := sup x i X x 1,, x n f 1 1 (x n ) f n (x n ) defines an n-norm on X Remark Through Proposition 5 we have shown that the dual of an n-normed space is also an n-normed space, whose n-norm is induced by the n-norm on the original space 3 Bounded n-linear Functional In this part we shall observe the notion of bounded n-linear functionals on n-normed spaces Let (X,,, ) be an n-normed space The function F : X n R is called an n-linear functional on X if F is linear in each variable The n-linear functional F is bounded if there exists k such that F (x 1,, x n ) k x 1,, x n, (x 1,, x n ) X n 4

5 If F is bounded, we define the norm of F by F (x 1,, x n ) F := sup x 1,,x n =0 x 1,, x n, or equivalently F := sup y 1,,y n =1 F (y 1,, y n ) The following fact gives two alternative formulas for F Fact 6 Let F be a bounded n-linear functional on X Then, we have F = inf{k : F (x 1,, x n ) k x 1,, x n, (x 1,, x n ) X n } = sup F (x 1,, x n ) x 1,,x n 1 Proof Let K = {k : F (x 1,, x n ) k x 1,, x n, (x 1,, x n ) X n } It is obvious that F K, and so inf K F Conversely, for each k K, we have F (x 1,, x n ) x 1,, x n whenever x 1,, x n = 0, so that F k But since this is true for all k K, we obtain F inf K Therefore F = inf K Next, if x 1,, x n 1, then This implies that Conversely, we have F = Therefore, F = k F (x 1,, x n ) F x 1,, x n F sup F (x 1,, x n ) F x 1,,x n 1 sup F (x 1,, x n ) sup F (x 1,, x n ) x 1,,x n =1 x 1,,x n 1 sup F (x 1,, x n ) x 1,,x n 1 To give an example, consider the n-normed space (R n,,, S ) with the standard basis {e 1,, e n } Define F by α 11 α 1n F (x 1,, x n ) := α n1 α = det [α ij] nn where x i = n j=1 α ije j, i = 1,, n Then, one may observe that F is a bounded n-linear functional on R n, with F x 1,, x n S Moreover, we have the following fact Fact 7 Let (X,, ) be a Hilbert space, which is also equipped with the standard n-norm,, S For fixed elements y 1,, y n in X, define F on X n by F (x 1,, x n ) = det [ x i, y j ] Then F is a bounded n-linear functional on X and F = y 1,, y n S 5

6 Proof Based on Fact 6, the norm of F is given by F = sup det [ xi, y j ] x 1,,x n S 1 The generalized Cauchy-Schwarz inequality gives us F Now, if we choose then we see that sup x 1,, x n S y 1,, y n S y 1,, y n S x 1,,x n S 1 x i = y i y1,, y n S, F (x 1,, x n ) = y 1,, y n S Therefore, we conclude that F = y 1,, y n S Note that, on the space l 2 (which is equipped with the n-norm,, 2 ), the above functional F in Fact 6 may be rewritten by the formula F (x 1,, x n ) := 1 det [x ijk ] det [y ijk ] j 1 j n where y 1,, y n are fixed in l 2 This formula also defines a bounded n-linear functional F on l p (1 p < ), which is equipped with the n-norm,, p, where x 1,, x n p := [ 1 det [x ijk ] p j 1 (see [6]) By Hölder s inequality, we have 1 det [x ijk ] det [y ijk ] x 1,, x n p y 1,, y n q j 1 j n for 1 p + 1 q = 1 Hence it follows that F is bounded, with F y 1,, y n q j n 4 Concluding Remark We have introduced a new formula of n-norm in a Hilbert space which can be viewed as a modification of Gähler s formula Accordingly, in a separable Hilbert space, we have four formulas of n-norms which are identical In addition, we have presented some results regarding the bounded n-linear functionals on n-normed spaces, especially on Hilbert spaces (equipped with the standard n-norm) In connection with the theory of n-normed spaces, we have introduced a formula for the induced n-norm on the dual space of an n-normed space The formula is defined in a natural way which is similar to that in the theory of normed spaces As a consequence of our result, we know that if X is an n-normed space, then the dual X is also an n-normed space It should be interesting to study the relation between an n-normed space and its dual, which is also an n-normed space 6 ] 1 p

7 References [1] S Gähler Lineare 2-normierte räume Math Nachr 28 (1964), 1 43 [2] S Gähler Untersuchungen über verallgemeinerte m-metrische räume I Math Nachr 40 (1969), [3] S Gähler Untersuchungen über verallgemeinerte m-metrische räume II Math Nachr 40 (1969), [4] S Gähler Untersuchungen über verallgemeinerte m-metrische räume III Math Nachr 41 (1970), [5] FR Gantmacher The Theory of Matrices AMS Chelsea Publishing Vol 1 (2000), [6] H Gunawan The space of p-summable sequences and its natural n-norm Bull Austral Math Soc 64 (2001), [7] H Gunawan On n-inner products, n-norms, and the Cauchy-Schwarz inequality Sci Math Jpn 55 (2002), [8] H Gunawan Inner products on n-inner product spaces Soochow J Math 28 (2002), [9] H Gunawan and Mashadi On n-normed spaces Int J Math Math Sci 27 (2001), [10] S Kurepa On the Buniakowsky-Cauchy-Schwarz inequality, Glas Mat III Ser 1 21 (1966), [11] A Misiak n-inner product spaces Math Nachr 140 (1989), [12] T Rangga Ruang Hasil Kali Dalam-n (in Indonesian) Undergraduate Thesis, ITB (2008) 1 Department of Mathematics, Bandung Institute of Technology, Bandung, Indonesia (Permanent address: Department of Mathematics Education, Universitas Pendidikan Indonesia, Bandung, Indonesia) sumanang@studentsitbacid 2 Department of Mathematics, Bandung Institute of Technology, Bandung, Indonesia hgunawan,oneswan@mathitbacid 7