TWO EQUIVALENT n-norms ON THE SPACE OF -SUMMABLE SEQUENCES RA Wibawa-Kusumah and H Gunawan 1 Abstract We rove the (strong) equivalence between two known n-norms on the sace l of -summable sequences (of real numbers) The first n-norm is derived from Gähler s formula [2], while the second is due to Gunawan [6] The equivalence is roved by using the roerties of the volume of n- dimensional aralleleieds in l 1 INTRODUCTION In the 1960 s, S Gähler [1, 2, 3, 4] develoed the theory of n-normed saces An n- norm on a real vector sace X (of dimension at least n) is a maing,, : X n R which satisfies the following four conditions: (N1) x 1,, x n = 0 if and only if x 1,, x n are linearly deendent; (N2) x 1,, x n is invariant under ermutation; (N3) αx 1,, x n = α x 1,, x n for α R; (N4) x 1 + x 1, x 2,, x n x 1, x 2,, x n + x 1, x 2,, x n The air (X,,, ) is called an n-normed sace Note that in this sace, we have x 1 + y, x 2,, x n = x 1, x 2,, x n for any y = c 2 x 2 + + c n x n See [7] for many other roerties of n-normed saces If X is a normed sace, then, according to Gähler, the following formula defines an n-norm on X: f 1 (x 1 ) f 1 (x n ) x 1,, x n := su f n (x 1 ) f n (x n ) f i X, f i 1 i = 1,,n Here X denotes the dual of X, which consists of bounded linear functionals on X For X = l (1 < ), the sace of -summable sequences (of real numbers), the above formula reduces to x1j z 1j x1j z nj x 1,, x n := su, z i l, z i 1 i = 1,,n xnj z 1j xnj z nj where denotes the usual norm on X = l and each of the sums is taken over j N Here denotes the dual exonent of, so that 1 + 1 = 1 2000 Mathematics Subject Classification: 46B05, 46B20, 46A45, 46A99, 46B99 Key words and Phrases: n-normed saces, -summable sequence saces, norm equivalence 1
2 RA Wibawa-Kusumah and H Gunawan In 2001, Gunawan [6] defined a different n-norm on l (1 < ) by x 1,, x n := 1 x 1j1 x nj1 abs n! j 1 j n x 1jn x njn where x i = (x ij ), i = 1,, n For = 2, this formula reduces to x 1, x 1 x 1, x n x 1,, x n 2 = x n, x 1 x n, x n where x i, x j denotes the usual inner roduct on l 2 Here x 1,, x n 2 reresents the volume of the n-dimensional aralleleied sanned by x 1,, x n in l 2 Thus, on l, we have two definitions of n-norms, one is derived from Gähler s formula and the other is due to Gunawan For = 2, one may verify that the two n- norms are identical (see [5]) The aim of this aer is to rove the (strong) equivalence between the two n-norms for 1 < We do so by invoking the volume formula of n-dimensional aralleleieds in l, which involves the notion of semi-inner roducts [8] This result solves the roblem osed in [10] 1/2, 1/, 2 MAIN RESULTS where On a normed sace (X, ), we may define the functional g : X 2 R by g (x, y) := x ( λ+ (x, y) + λ (x, y) ), 2 λ ± (x, y) := lim t ±0 The functional g satisfies the following roerties: (F1) g (x, x) = x 2 for every x X; x + ty x t (F2) g (αx, βy) = αβg (x, y) for every x, y X, α, β R; (F3) g (x, x + y) = x 2 + g (x, y) for every x, y X; (F4) g (x, y) x y for every x, y X If, in addition, the functional g (x, y) is linear with resect to y X, then g is called a semi inner roduct on X For examle, for 1 <, the functional g (x, y) := x 2 x j 1 sgn (x j ) y j, x = (x j ), y = (y j ) l, (1) j defines a semi inner roduct on l, where is the usual norm on l By using the semi inner roduct g, we can define an orthogonality relation on X by x g y g (x, y) = 0
Two Equivalent n-norms on the Sace of -Summable Sequences 3 In general, x g y does not imly y g x, since g is not always commutative Next, we can define the g-orthogonal rojection of y on x by y x := g (x, y) x 2 x, and obtain the g-orthogonal comlement y y x Note here that x g y y x Moreover, the g-orthogonal rojection of y on S = san {x 1,, x k } with Γ(x 1,, x k ) := det[g(x i, x j )] 0 is given by 0 x 1 x k 1 g (x 1, y) g (x 1, x 1 ) g (x 1, x k ) y S := Γ (x 1,, x k ), g (x k, y) g (x k, x 1 ) g (x k, x k ) and the g-orthogonal comlement y y S is given by y x 1 x k 1 g (x 1, y) g (x 1, x 1 ) g (x 1, x k ) y y S := Γ (x 1,, x k ) g (x k, y) g (x k, x 1 ) g (x k, x k ) Observe here that x i g y y S for i = 1,, k Now, let {x 1,, x n } be a linearly indeendent set of vectors in X Then, as in [9], we can obtain the left g-orthogonal sequence x 1,, x n through the following rocedure: x 1 = x 1 and x i = x i (x i ) Si 1, where S i 1 = san { x 1,, xi 1} for i = 2,, n Note here that xi g x j whenever i < j Next, as in [8], we can define the volume of the n-dimensional aralleleieds sanned by x 1,, x n by n V (x 1,, x n ) := x i (2) Since g may not be commutative, the value of V (x 1,, x n ) may not be invariant under ermutation of (x 1,, x n ) If x 1,, x n are linearly deendent, then we shall define V (x 1,, x n ) = 0 The following theorem gives an estimate for the volume of an n-dimensional aralleleieds in l in terms of Gunawan s n-norm Theorem 21 Let {x 1,, x n } be any set in l Then we have (n!) 1/ 1 x 1,, x n V (x i1,, x in ) (n!) 1/ x 1,, x n for any ermutation (i 1,, i n ) of (1,, n) Proof The uer estimate is already roved in [8] Now, to rove the lower estimate, it suffices to show (n!) 1/ 1 x 1,, x n V (x 1,, x n ) because x 1,, x n is invariant under ermutation of (x 1,, x n ) Assuming that x 1,, x n are linearly indeendent, let x 1,, x n be the left g-orthogonal sequence obtained from x 1,, x n Then, by basic roerties of an n-norm, we have i=1 x 1,, x n = x 1,, x n
4 RA Wibawa-Kusumah and H Gunawan Next (see [6], Fact 31), we have x 1,, x n (n!) 1/ n i=1 x i where 1 + 1 = 1 It thus follows that n (n!) 1/ 1 x 1,, x n x i = V (x 1,, x n ), i=1 as desired The following theorem gives an estimate for the volume in terms of Gähler s n-norm Theorem 22 Let {x 1,, x n } be any set in l Then, we have (n!) 1 x 1,, x n V (x i1,, x in ) x 1,, x n for any ermutation (i 1,, i n ) of (1,, n) Proof As before, it suffices to show (n!) 1 x 1,, x n V (x 1,, x n ) x 1,, x n for any linearly indeendent set {x 1,, x n } in l The lower estimate follows from the inequality x 1,, x n (n!) 1/ x 1,, x n (see [10], Fact 25) and the lower estimate in Theorem 21 Next, to rove the uer estimate, let x 1,, x n be the left g-orthogonal sequence obtained from x 1,, x n Recall that x 1j z 1j x 1j z nj x 1,, x n = su z i l, z i 1 i=1,,n x nj z 1j x nj z nj For each i = 1,, n, take z i = (z ij ) where z ij := x i 1 x ij 1 sgn(x ij ) We observe that z i = 1, and hence x 1 1 g(x 1, x 1) x n 1 g(x n, x 1) x 1,, x n, x 1 1 g(x 1, x n) x n 1 g(x n, x n) where g is the functional defined by the formula (1) Here g(x i, x j ) = 0 for i < j and g(x i, x i ) = x i 2 for i = 1,, n, and so the determinant on the right hand side is equal to n i=1 x i = V (x 1,, x n ) Meanwhile, the left hand side is equal to x 1,, x n Therefore, we obtain x 1,, x n V (x 1,, x n ), which is what we need to rove As a consequence of Theorems 21 and 22, we get the following result, which tells us that Gunawan s and Gähler s n-norms on l are (strongly) equivalent
Two Equivalent n-norms on the Sace of -Summable Sequences 5 Theorem 23 For any x 1,, x n l, we have (n!) 1/ 1 x 1,, x n x 1,, x n (n!) 1/ x 1,, x n Note The uer estimate follows from [10], Fact 25 3 CONCLUDING REMARKS We have just seen that Gähler s and Gunawan s n-norms on l are (strongly) equivalent While Gähler s formula uses the functionals on l as a normed sace, Gunawan s uses the Plücker coordinates of the vectors (thanks to Norman Wildberger who brought these coordinates into our attention) In this case, Gunawan s formula allows us to comute the value of the n-norm x 1,, x n directly from the coordinates of the vectors x 1,, x n If one insists to involve functionals on l, one may actually use the fact that the dual sace l is also an n-normed sace, as well as l (1 < ) Thus, one may define the following n-norm on l : x1j z 1j x1j z nj x 1,, x n := su z i l, z 1,,z n 1, xnj z 1j xnj z nj where 1 + 1 = 1 Here,, is Gunawan s n-norm on l For = 2, we observe that x 1,, x n 2 = x 1,, x n 2, so that we have the duality x 1,, x n 2 := su z i l 2, z 1,,z n 2 1 x1j z 1j x1j z nj xnj z 1j xnj z nj For other values of, one may show that the above n-norm is at least equivalent to Gunawan s (and hence it is also equivalent to Gähler s) See [5] for related results Acknowledgement The research is suorted by ITB Research and Innovation Program 2011 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), 165-189 3 S Gähler, Untersuchungen über verallgemeinerte m-metrische Räume II, Math Nachr 40 (1969), 229 264 4 S Gähler, Untersuchungen über verallgemeinerte m-metrische Räume III, Math Nachr 41 (1970), 23 26 5 SM Gozali, H Gunawan and O Neswan, On n-norms and bounded n-linear functionals in a Hilbert sace, Ann Funct Anal 1 (2010), 72 79
6 RA Wibawa-Kusumah and H Gunawan 6 H Gunawan, The sace of -summable sequences and its natural n-norms, Bull Austral Math Soc 64 (2001), 137 147 7 H Gunawan and Mashadi, On n-normed saces, Int J Math Math Sci 27 (2001), 631 639 8 H Gunawan, W Setya-Budhi, Mashadi and S Gemawati, On volume of n-dimensional aralleleieds in l saces, Univ Beograd Publ Elektrotehn Fak Ser Mat 16 (2005), 48 54 9 PM Miličić, On the Gram-Scmidt rojection in normed saces, Univ Beograd Publ Elektrotehn Fak Ser Mat 4 (1993), 89 96 10 A Mutaqin and H Gunawan, Equivalence of n-norms on the sace of -summable sequences, J Indones Math Soc 16 (2010) 1 Deartment of Mathematics, Institut Teknologi Bandung, Bandung, Indonesia E-mail: hgunawan@mathitbacid