J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae7.
EQUIVALENCE OF n-NORMS ON THE SPACE OF p-SUMMABLE SEQUENCES Anwar Mutaqin1 and Hendra Gunawan2 Department of Mathematics Education.
Universitas Sultan Ageng Tirtayasa.
Serang.
Indonesia, anwarmutaqin@gmail.
Department of Mathematics.
Institut Teknologi Bandung.
Bandung.
Indonesia, hgunawan@math.
Abstract.
We study the relation between two known n-norms on `p , the space of p-summable sequences.
One n-norm is derived from GaOhlerAos formula .
, while the other is due to Gunawan .
We show in particular that the convergence in one n-norm implies that in the other.
The key is to show that the convergence in each of these n-norms is equivalent to that in the usual norm on `p .
Key words: n-normed spaces, p-summable sequence spaces, n-norm equivalence.
Abstrak.
Dalam makalah ini dipelajari kaitan antara dua norm-n di `p , ruang barisan summable-p.
Norm-n pertama diperoleh dari rumus GaOhler .
, sementara norm-n kedua diperkenalkan oleh Gunawan .
Ditunjukkan antara lain bahwa kekonvergenan dalam norm-n yang satu mengakibatkan kekonvergenan dalam normn lainnya.
Kuncinya adalah bahwa kekonvergenan dalam masing-masing norm-n tersebut setara dengan kekonvergenan dalam norm biasa di `p .
Kata kunci: ruang norm-n, ruang barisan summable-p, kesetaraan norm-n Introduction In .
Gunawan introduced an n-norm on `p .
O p O O), the space of p-summable sequences .
f real number.
, given by the formula A x1j1 A A A xnj1 Ap 1/p 1 kx1 , .
, xn kp := a abs A .
n! j A x1j A A A xnj A 2000 Mathematics Subject Classification: 46B15 Received: 31-03-2010, accepted: 05-08-2010.
Mutaqin and H.
Gunawan for 1 O p < O, and A x1j1 kx1 , .
, xn kO = sup sup A A A sup abs A .
j1 j2
A xnj
a a A x1jn AA
A ,
xnj A where xi = .
ij ), i = 1, .
, n.
For p = 2, the above formula may be rewritten as A hx1 , x1 i A A A hx1 , xn i A1/2 kx1 , .
, xn k2 = A A .
A hxn , x1 i A A A hxn , xn i A where hxi , xj i denotes the usual inner product on `2 .
Here kx1 , .
, xn k2 represents the volume of the n-dimensional parallelepiped spanned by x1 , .
, xn in `2 .
In general, an n-norm on a real vector space X is a mapping kA, .
Ak : X n Ie R which satisfies the following four conditions:
(N.
kx1 , .
, xn k = 0 if and only if x1 , .
, xn are linearly dependent.
(N.
kx1 , .
, xn k is invariant under permutation.
(N.
kx1 , .
, xn k = || kx1 , .
, xn k for OO R.
(N.
kx1 x01 , x2 , .
, xn k O kx1 , x2 , .
, xn k kx01 , x2 , .
, xn k.
The theory of n-normed spaces was developed by GaOhler in 1969 and 1970 .
, 4, .
The special case where n = 2 was studied earlier, also by GaOhler, in 1964 .
Related work may be found in .
For more recent works, see .
, 8, .
If X is equipped with a norm k A k, then according to GaOhler, one may define an n-norm on X .
ssuming that X is at least n-dimensiona.
by the formula A f1 .
1 ) A A A f1 .
n ) A O kx1 , .
, xn k := A.
fi OOX , kfi kO1 A fn .
1 ) A A A fn .
n ) A i = 1,.
Here X 0 denotes the dual of X, which consists of bounded linear functionals on X.
For X = `p .
O p < O), we know that X 0 = `p with p1 p10 = 1.
In this case the above formula reduces to A P x1j z1j A A A x1j znj AA kx1 , .
, xn kp := A, zi OO`p , kzi kp0 O1 A xnj z1j A A A xnj znj A i = 1,.
where kAkp0 denotes the usual norm on `p and each of the sums is taken over j OO N.
Thus, on `p , we have two definitions of n-norms, one is due to Gunawan and the other is derived from GaOhlerAos formula.
For p = 2, one may verify that the two n-norms are identical.
The purpose of this paper is to study the relation between the two n-norms on `p for 1 O p < O.
In particular, we shall show that the two n-norms are weakly equivalent, that is, the convergence in one n-norm implies that in the other.
Here Equivalence of n-Norms on the Space of p-Summable Sequences a sequence .
) in an n-normed space (X, kA, .
is said to converge to x OO X if kx.
Oe x, x2 , .
, xn k Ie 0 as m Ie O, for every x2 , .
, xn OO X.
For convenience, we prove the result for n = 2 first, and then extend it to any n Ou 2.
Main Results Recall that GunawanAos definition of 2-norm on `p .
O p O O) is given by A x
1 XX
kx, ykp = abs AA j Ap 1/p xk AA yk A A x kx, ykO = sup sup abs AA j xk AA yk A if 1 O p < O, and Meanwhile.
GaOhlerAos definition is given by kx, ykOp = A P A P xj zj yj zj z,wOO`p , kzk 0 , kwk 0 O1 P xj wj A .
yj wj A By the same trick as in .
, one may obtain kx, ykOp = 1 X X AA xj A yj z,wOO`p0 , kzkp0 , kwkp0 O1 2 j xk AA AA zj yk A A wj zk AA wk A From the last expression, we have the following fact.
Fact 2.
The inequality kx, ykOp O 21/p kx, ykp holds for every x, y OO `p .
Proof.
By HoOlderAos inequality for p1 p10 = 1, we have 1 X X AA xj
A yj
xk AA AA zj
yk A A wj
A xj
zk A 1 X X
A yj
1 XX
Ap 1/p xk A yk A A z abs AA j Ap0 1/p zk A wk A A.
Mutaqin and H.
Gunawan Now, observe that A z abs AA j 1/p0 0 Ap0 1/p X XA Ap0 zk AA .
j wk | .
k wj | O wk A 1/p0 1/p0 O .
j wk .
k wj .
= 2 kzkp0 kwkp0 .
But for kzkp0 , kwkp0 O 1 we have A z abs AA j Ap0 1/p zk AA O 21Oe.
/p ) = 21/p .
wk A This proves the inequality.
Note that for p = 1.
HoOlderAos inequality gives AA 1 X X AA xj xk AA AA zj zk AA A yj yk A A wj wk A O kx, yk1 .
kz, wkO .
But kz, wkO O 2 kzkO kwkO .
), and so taking the supremum over kzkO and kwkO O 1, we get kx, ykO1 O 2kx, yk1 .
Corollary 2.
2 If .
) converges in kA.
Akp , then it also converges .
o the same limi.
in kA.
AkOp .
We shall show next that the convergence in kA.
AkOp also implies the convergence in kA.
Akp .
We do so by showing that: .
the convergence in kA.
AkOp implies that in k A kp , and .
the convergence in k A kp implies that in kA.
Akp .
The second implication is already proved in .
sing the inequality kx, ykp O 21Oe.
kxkp kykp ).
Hence it remains only to show the first implication.
Theorem 2.
3 If .
) converges in kA.
AkOp , then it also converges .
o the same limi.
in k A kp .
Proof.
Let .
) be a sequence in `p which converges to x OO `p in kA.
AkOp .
Then, for any A > 0, there exists an N OO N such that for m Ou N we have AA 1 X X AA xj .
Oe xj xk .
Oe xk AA AA zj zk AA A A wj wk A < A for every y OO `p and z, w OO `p with kzkp0 , kwkp0 O 1.
[Notice here that, for each m, we have x.
= .
) OO `p .
] In particular, if we take y := .
, 0, 0, .
), z = .
j ) Equivalence of n-Norms on the Space of p-Summable Sequences with zj := sgn.
Oexj ).
Oexj .
Oe1 pOe1 kx.
Oexkp and w := .
, 0, 0, .
), then we have O .
Oe xj .
Oe xkpOe1 < A.
[Here we are handling only the case where kx.
Oe xkp 6= 0.
] Next, if we take 1 ).
Oex1 | y := .
, 1, 0, .
), z = .
1 , 0, 0, .
) with z1 := sgn.
Oex kx.
OexkpOe1 .
, 1, 0, .
), then we have pOe1 and w := .
Oe x1 .
< A.
Oe xkpOe1 Adding up, we get kx.
Oe xkp = O .
Oe xj .
Oe xkpOe1 This shows that .
) converges to x in k A kp .
< 2A.
Corollary 2.
4 A sequence is convergent in kA.
AkOp if and only if it is convergent .
o the same limi.
in kA.
Akp .
All these results can be extended to n-normed spaces for any n Ou 2.
As an extension of Fact 2.
1, we have:
Fact 2.
5 The inequality kx1 , .
, xn kOp O .
!)1/p kx1 , .
, xn kp holds for every x1 , .
xn OO `p .
Corollary 2.
6 If .
) converges in kA, .
Akp , then it converges .
o the same limi.
in kA, .
AkOp .
Analogous to Theorem 2.
3, we have:
Theorem 2.
7 If .
) converges in kA, .
AkOp , then it also converges .
o the same limi.
in k A kp .
Proof.
Let .
) be a sequence in `p which converges to x1 = .
11 , x12 , .
) OO `p O in kA, .
Akp .
Then, for any A > 0, there exists an N OO N such that for m Ou N we AA A x1j1 .
Oe x1j1 A A A x1jn .
Oe x1jn A A z1j1 A A A z1jn A A A .
A < A
a AA AA .
n! j jn A a znj1 A A A znjn A for every x2 , .
, xn OO `p and z1 , .
, zn OO `p with kz1 k , .
, kzn k O 1.
Now, take xk = zk := .
, .
, 0, 1, 0, .
) for every k = 2, .
, n, where 1 is .
1 Oe .
-th A.
Mutaqin and H.
Gunawan term and z1 = .
11 , z12 , .
) OO `p with z1j := we have sgn.
Oex1j ).
Oex1j .
Oe1 , then kx1 .
Oex1 kpOe1 Oe x1j | pOe1 < A.
j1 =n kx1 .
Oe x1 kp Next, if we take xk = zk := .
, .
, 0, 1, 0, .
) for every k = 2, .
, n, where 1 is pOe1 11 ).
Oex11 | k-th term, and z1 := .
11 , 0, 0, .
) with z11 := sgn.
Oex , then kx .
Oex kpOe1 we have .
Oe x11 | pOe1 < A.
Oe x1 kp Similarly, if we alter the position of the entry 1 in xk and zk for k = 2, .
, n, and change the nonzero entry of z1 accordingly, then we can get .
Oe x12 | pOe1 < A kx1 .
Oe x1 kp and so on until Ax1.
Oe.
Oe x1.
Oe.
Ap pOe1 kx1 .
Oe x1 kp < A.
Adding up, we get kx1 .
Oe x1 kp = O .
Oe x1j | pOe1 < nA.
j1 =1 kx1 .
Oe x1 kp This shows that .
) converges to x in k A kp .
Corollary 2.
8 A sequence is convergent in kA, .
AkOp if and only if it is convergent .
o the same limi.
in kA, .
Akp .
Related to the above results, one may also prove that a sequence is Cauchy in kA, .
AkOp if and only if it is Cauchy in kA, .
Akp .
[A sequence .
) in an n-normed space (X, kA, .
is Cauchy if given A > 0 there exists an N OO N such that kx.
Oe x.
, x2 , .
, xn k < A whenever l, m Ou N , for every x2 , .
, xn OO X.
Since (`p , kA, .
Akp ) is a Banach space .
, we conclude, by Theorem 2.
7, that (`p , kA, .
AkOp ) also forms an n-Banach space.
Concluding Remarks As we have mentioned earlier, the case where p = 2 is of course special.
Here, the two n-norms kA, .
Ak2 and kA, .
AkO2 are identical.
Indeed, by using Cauchy-Schwarz inequality .
), one may obtain A hx1 , z1 i A A A hx1 , zn i A kx1 , .
, xn kO2 = A O kx1 , .
, xn k2 .
zi OO` , kzi k2 O1 A hxn , z1 i A A A hxn , zn i A i = 1,.
Equivalence of n-Norms on the Space of p-Summable Sequences By taking z1 , .
, zn to be the orthonormalized vectors obtained from x1 , .
, xn through Gram-Schmidt process, one can show that the above upper bound is actually attained.
Hence we have kx1 , .
, xn kO2 = kx1 , .
, xn k2 .
For p 6= 2, things are not so simple and we have difficulties in proving the strong equivalence between the two n-norms kA, .
AkOp and kA, .
Akp .
The research on this problem, however, is still ongoing.
Acknowledgement.
The research was carried out while the first author did his master thesis at Faculty of Mathematics and Natural Sciences.
Institut Teknologi Bandung.
References