Strongly Summable Sequences Defined Over Real n -Normed Spaces ∗
Hemen Dutta
†, B. Surender Reddy
‡, Sui Sun Cheng
§Received 11 November 2008
Abstract
The main aim of this article is to study subsets of real linear n-normed spaces consisting of strongly Ces`aro summable and strongly lacunary summable sequences. Some standard facts as linearity, existence of norms and completeness with respect to these norms are investigated. Also some facts on equivalence of various norms on such constructed Banach spaces are presented, and we show that their topology can be fully described by using derived norm (norm). Further we investigate the relationship between the spaces and provide some examples and possible applications.
1 Introduction
Different types of complex sequences of the formx={xk}∞k=1 or in short (xk),under various norms have been studied to great extent. In particular, the linear spacewof all complex sequences (xk) endowed with the usual operations and the supremum norm kxk∞= supk|xk|,as well as its subspaces `∞, candc0,consisting respectively of all, bounded, convergent and null sequences, are well studied.
The standard concept of a norm has, however, been extended. Therefore, the space w under these new norms may be of interests in various applications. In this paper, we intend to study the properties of several subsets of the linear space wunder the so called n-norms.
Let us first recall the concept of an n-norm. Let n ∈ N and X be a real linear space of dimension d≥n≥2. A real valued function k., . . . , .k :Xn →R satisfying the following four properties:
(N1) kx1, x2, . . . , xnk= 0 if and only ifx1, x2, . . . , xnare linearly dependent vectors, (N2) kx1, x2, . . . , xnk = kxj1, xj2, . . . , xjnk for every permutation (j1, j2, . . . , jn) of
(1,2, . . . , n),
(N3) kαx1, x2, . . . , xnk=|α|kx1, x2, . . . , xnkfor allα∈R,
∗Mathematics Subject Classifications: 46B04, 40A05, 46A45.
†Department of Mathematics, Gauhati University, Kokrajhar Campus, Assam, India
‡Department of Mathematics, PGCS, Saifabad, Osmania University, Hyderabad-500004, AP, India
§Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.
199
(N4) kx+x0, x2, . . . , xnk ≤ kx, x2, . . . , xnk+kx0, x2, . . . , xnk for allx, x0, x2, . . . , xn∈ X,
is called ann-norm on X and the pair (X,k., . . . , .k) is called a linearn-normed space.
The concept of a 2-normed space was developed by G¨ahler [3] in the mid of 1960’s, while that of ann-normed space can be found in Misiak [12]. Since then, many others have studied this concept and obtained various results; see for instance Gunawan [6, 7]
and Gunawan and Mashadi [8, 9].
A trivial example of an n-normed space is X = Rn equipped with the following Euclidean n-norm:
kx1, x2, . . . , xnkE=|det(xij)| where xi= (xi1, . . . , xin)∈Rn for each i= 1,2, . . . , n.
If (X,k., . . . , .k) is ann-normed space of dimensiond≥n≥2 and{a1, a2, . . . , an}a linearly independent set inX, then the following functionk., . . . , .k∞onXn−1 defined by
kx1, x2, . . . , xn−1k∞= max{kx1, x2, . . . , xn−1, aik:i= 1,2, . . . , n}
defines an (n−1)-norm onX with respect to{a1, a2, . . . , an} and this is known as a derived (n−1)-norm onX.
The standard n-norm on X, where X is a real inner product space of dimension d≥n,is defined as
kx1, x2, . . . , xnkS =
hx1, x1i . . . hx1, xni ... . .. ... hxn, x1i . . . hxn, xni
12
,
whereh., .idenotes the inner product onX. IfX=Rn, then thisn-norm is exactly the same as the Euclidean n-norm k., . . . , .kE mentioned earlier. Forn= 1, this n-norm reduces to the usual normkx1k=hx1, x1i12.
A sequence (xk) in ann-normed space (X,k., . . . , .k) is said toconvergeto someL∈ X in the n-norm if limk→∞kxk−L, w2, w3. . . , wnk = 0 for everyw2, w3. . . , wn ∈X. A sequence (xk) in ann-normed space (X,k., . . . , .k) is said to beCauchywith respect to then-norm if limk,l→∞kxk−xl, w2, w3. . . , wnk= 0 for everyw2, w3. . . , wn∈X. If every Cauchy sequence inX converges to someL∈X, thenX is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.
Now we state the following three useful results as Lemmas which can be found in [9].
LEMMA 1. Everyn-normed space is an (n−r)-normed space for allr= 1,2, . . . ,n− 1. In particular, every n-normed space is a normed space.
LEMMA 2. A standard n-normed space is complete if and only if it is complete with respect to the usual normk.kS =h., .i1/2.
LEMMA 3. On a standardn-normed spaceX, the derived (n−1)-normk., . . . , .k∞, defined with respect to orthonormal set {e1, e2, . . . , en}, is equivalent to the standard
(n−1)-norm k., . . . , .kS. Precisely, we have
kx1, x2, . . . , xn−1k∞≤ kx1, x2, . . . , xn−1kS ≤√
nkx1, x2, . . . , xn−1k∞ for allx1, x2, . . . , xn−1, where
kx1, x2, . . . , xn−1k∞= max{kx1, x2, . . . , xn−1, eikS :i= 1,2, . . . , n}.
Next we recall two subsets of the space w. The first is the space |σ1| of strongly Ces`aro summable sequences (see e.g. Borwein [1], Freedman, Sember and Raphael [2]
and Maddox [11]). It is defined as
|σ1|= (
x= (xk) : there existsLsuch that lim
p
1 p
p
X
k=1
|xk−L|= 0 )
and it is a Banach space normed by kxk= sup
p
1 p
p
X
k=1
|xk|
! .
Next, by a lacunary sequenceθ= (kp); we mean an increasing sequence of non-negative integers with hp = (kp−kp−1) → ∞, where k0 = 0, as p → ∞. We denote Ip = (kp−1, kp] andηp= kkp
p−1 forp= 1,2,3, . . .. The space of strongly lacunary summable sequence Nθ was defined by Freedman, Sember and Raphael [2] as follows:
Nθ =
x= (xk) : lim
p→∞
1 hp
X
k∈Ip
|xk−L|= 0, for some L
.
The space Nθ is a Banach space with the norm kxkθ= sup
p
1 hp
X
k∈Ip
|xk|
.
Throughout the article (X,k., . . . , .kX) will be an n-normed space andw(X) will denote X-valued sequence space. The n-norm k., . . . , .kX on X is either a standard n-norm or non-standardn-norm. In general, we writek., . . . , .kXand for standard case we writek., . . . , .kS. Again for derived norms we usek., . . . , .k∞.
2 Main Results
In this section we extend the notion of strongly Ces`aro summable sequences and strongly lacunary summable sequences ton-normed space valued sequences.
We denote by|σ1|(X) the set of allX-valued strongly Ces`aro summable sequences defined as the set of allx∈w(X) such that
p→∞lim 1 p
p
X
k=1
kxk−L, z1, ..., zn−1kX = 0 for everyz1, ..., zn−1∈X and for someL.
ForL= 0, we write this corresponding space as |σ1|0(X).
Letθ be a lacunary sequence. Then we denote by Nθ(X) the set of all X-valued strongly lacunary summable sequences x∈w(X) such that
p→∞lim 1 hp
X
k∈Ip
kxk−L, z1, ..., zn−1kX = 0 for everyz1, ..., zn−1∈X and for someL.
ForL= 0, we write this space asNθ0(X).
In the special case whereθ= (2p), we have Nθ(X) =|σ1|(X).
THEOREM 1. The following are true:
(i) IfX is ann-Banach space then|σ1|(X) is a Banach space normed by kxk= sup
p
1 p
p
X
k=1
kxk, z1, ..., zn−1kX
!
. (1)
(ii) IfX is an n-Banach space then Nθ(X) is a Banach space normed by kxkθ= sup
p
1 hp
X
k∈Ip
kxk, z1, ..., zn−1kX
. (2) PROOF. It is easy to see that|σ1|(X) is a normed linear space. To prove complete- ness, let (xi) be a Cauchy sequence in|σ1|(X), wherexi= (xik) = (xi1, xi2, ...) for each i∈N. Then for a given ε >0, there exists a positive integern0such that
kxi−xjk= sup
p
1 p
p
X
k=1
kxik−xjk, z1, ..., zn−1kX
!
< ε, for alli, j≥n0. It follows that
1 p
p
X
k=1
kxik−xjk, z1, ..., zn−1kX< ε, for alli, j≥n0and for allp≥1.
Hence (xik) is a Cauchy sequence in X for allk ∈N. SinceX is an n-Banach space, (xik) is convergent inX for allk∈N. For simplicity, let lim
i→∞xik=xk (say), exists for each k∈N. Now we can find that
j→∞lim 1 p
p
X
k=1
kxik−xjk, z1, ..., zn−1kX < ε, for alli≥n0and for allp≥1.
Thus,
sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−1kX
!
< ε, for alli≥n0.
It follows that (xi−x)∈ |σ1|(X). Since (xi)∈ |σ1|(X) and |σ1|(X) is a linear space, so we havex=xi−(xi−x)∈ |σ1|(X). This completes the proof of part (i).
The proof of (ii) similar and skipped. The proof is complete.
COROLLARY 2. LetX be equipped with the standardn-norm. Then (i) ifX is a Banach space then|σ1|(X) is a Banach space normed by
kxk= sup
p
1 p
p
X
k=1
kxk, z1, ..., zn−1kX
! ,
(ii) ifX is a Banach space then Nθ(X) is a Banach space normed by kxkθ= sup
r
1 hr
X
k∈Ir
kxk, z1, ..., zn−1kX
!
Indeed, the proof follows by combining Lemma 2 and Theorem 1, and is skipped.
We now use the notion derived norms to define some other norms on the spaces and investigate the relationship among these norms.
Let{a1, a2, ..., an}be a linearly independent set inX. Then
kxk, z1, ..., zn−r−1k∞= max{kxk, z1, ..., zn−r−1, ai1, ai2, ..., airkX}, {i1, ..., ir} ⊆ {1, ..., n} is an derived (n−r)-norm onX for eachr= 1,2, ..., n−1 and for each k≥1. Hence we have the following Theorem.
THEOREM 3. Let{a1, a2,· · ·, an}be a linearly independent set inX. Then (i) |σ1|(X) is a normed linear space, with normk.krdefined by
kxkr= sup
p
1 p
p
X
k=1
kxk, z1, ..., zn−r−1k∞
!
for eachr= 1,2, ..., n−1, (3) (ii)Nθ(X) is a normed linear space, with norm k.krθ defined by
kxkrθ= sup
r
1 hr
X
k∈Ir
kxk, z1, ..., zn−r−1k∞
!
for each r= 1,2, ..., n−1. (4) We call the above norms as the derived (n−r)-norm for eachr= 1,2, ..., n−1.
Proof is a routine verification and so omitted.
THEOREM 4. IfX is an (n−r)-Banach spaces for each r = 1,2, ..., n−1, then
|σ1|(X) is a Banach with normk.krdefined by (3) and Nθ(X) is a Banach space with norm k.krθdefined by (4).
Proof is same with the proof of Theorem 1 and is omitted.
THEOREM 5. If (xi) converges to anxin|σ1|(X) in the normk.k defined by (1), then (xi) also converges toxin the normk.krdefined by (3) forr= 1.
PROOF. Let (xi) converges to xin|σ1|(X) in the norm k.k.Thenkxi−xk −→0 as i−→ ∞. Using the definition of norm (1), we get
sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−1kX
!
−→0 asi−→ ∞.
Let{a1, a2,· · ·, an} be any linearly independent set inX. Then sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−2, ajkX
!
−→0 asi−→ ∞ for eachj= 1,2, ..., n.
Hence
sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−2k∞
!
−→0 asi−→ ∞.
Thuskxi−xk1−→0 asi−→ ∞. Hence (xi) converges toxin the normk.k1. IfXis equipped with the standardn-norm and derived norm onX are with respect to an orthonormal set then the converse of the above Theorem is also true. Conse- quently we have the following Theorem.
THEOREM 6. LetX be a standardn-normed space and the derived (n−1)-norm onX is with respect to an orthonormal set. Then (xi) is convergent in|σ1|(X) in the norm k.k defined by (1), if and only if (xi) is convergent in|σ1|(X) in the normk.kr defined by (3) forr= 1.
PROOF. In view of the above Theorem it is enough to prove that (xi) is convergent in the normk.k1implies (xi) is convergent in the normk.k. Let (xi) is converges tox in|σ1|(X) in the normk.k1. Then kxi−xk1−→0 as i−→ ∞. Using (3) withr= 1, we get
sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−2k∞
!
−→0 asi−→ ∞.
Now one can observe that kxik−xk, z1,· · ·, zn−1kS ≤ kxik−x, z1, ..., zn−2kSkzn−1kS, where k., . . . , .kS and k.kS on the right hand side denote the standard (n−1)-norm and the usual norm on X respectively. Since the derived (n−1)-norm on X is with respect to an orthonormal set, using Lemma 3, we have
kxik−xk, z1, ..., zn−1kS≤√
nkxik−x, z1, ..., zn−2k∞kzn−1kS,
and in this case k., . . . , .k∞ on the right hand side is the derived (n−1)-norm which we used to define the norm k.k1. Therefore
sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−1kS
!
≤sup
p
1 p
p
X
k=1
√n
xik−xk, z1, ..., zn−2
∞kzn−1kS .
Hence
sup
p
1 p
p
X
k=1
kxik−xk, z1, ..., zn−1kS
!
−→0 asi−→ ∞.
Thuskxi−xk −→0 asi−→ ∞. That is, (xi) is converges toxin|σ1|(X) in the norm k.k. The proof is complete.
COROLLARY 7. Let X be a standard n-normed space and the derived (n−r)- norms on X are with respect to an orthonormal set. Then a sequence in |σ1|(X) is
convergent in the norm k.k defined by (1) if and only if it is convergent in the norm k.k1 and, by induction, in the norm k.kr defined by (3) for all r = 1,2, ..., n−1. In particular, a sequence in |σ1|(X) is convergent in the norm k.k if and only if it is convergent in the normk.kn−1defined by
kxkn−1= sup
p
1 p
p
X
k=1
kxkk∞
!
. (5)
THEOREM 8. LetX be a standardn-normed space and the derived (n−r)-norms on X for all r= 1,2, ..., n−1 are with respect to an orthonormal set. Then |σ1|(X) is complete with respect to the norm k.k defined by (1) if and only if it is complete with respect to the normk.k1 defined by (3). By induction,|σ1|(X) is complete with respect to the norm k.k if and only if it is complete with respect to the normk.kn−1 defined by (5).
PROOF. By replacing the phrases ‘(xi) converges tox’ with ‘(xi) is Cauchy’ and
‘xi−x’ with ‘xi−xj’, we see that the analogues of Theorem 5, Theorem 6 and Corollary 7 hold for Cauchy sequences. This completes the proof.
REMARK 1. It we replace the space |σ1|(X) byNθ(X), analogues of Theorem 5, Theorem 6, Corollary 7 and Theorem 8 hold forNθ(X).
EXAMPLE 4. Let us takeX =R3 and consider a 3-normk., ., .kXdefined as:
kx1, x2, x3kX =|det(xij)|,
where xi = (xi1, xi2, xi3) ∈ R3 for each i = 1,2,3. Consider the divergent sequence x ={0,1,0,1,0,· · · } ∈ w(X), where k = (k, k, k) for each k = 0,1. Let us consider a basis {e1, e2, e3} ofX =R3, where e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1). Now kxkk∞ = max{kxk, ei1, ei2kX}, {i1, i2} ⊆ {1,2,3}is an derived norm onX. Thenx belong toNθ(X) and|σ1|(X), forθ= (2p). Here actually strongly Ces`aro summability method transform the sequence x into the sequence y, where y =
0,12,13,12,25,· · · , which converges to 12. In other words we can say that xhas the generalized limit 12. Hence xis a 3-nls valued strongly Ces`aro summable sequence.
REMARK 2. Associated to the derived normk.kn−1, we can define balls (open) S(x, ε) centered atxand radiusεas follows:
S(x, ε) ={y:kx−ykn−1< ε}. Using these balls, Theorem 8 becomes:
THEOREM 9. A sequence (xk) is convergent toxin|σ1|(X) if and only if for every ε >0, there existsn0∈N such thatxk∈S(x, ε) for allk≥n0.
Hence we have the following important result.
THEOREM 10. A space |σ1|(X) is a normed space and its topology agrees with that generated by the derived normk.kn−1.
Our next aim is to investigate the relationship among the spaces|σ1|(X) andNθ(X).
PROPOSITION 11. Letθ= (kp) be a lacunary sequence with lim infpηp>1, then
|σ1|(X)⊆Nθ(X).
PROOF. Let lim infηp>1. Then there exists a ν >0 such that 1 +ν ≤ηpfor all p≥1. Letx∈ |σ1|(X). Then there exists someL∈X such that
t→∞lim 1 t
t
X
k=1
kxk−L, z1, ..., zn−1kX= 0, for everyz1, ..., zn−1∈X.
Now we write 1 hp
X
k∈Ip
kxk−L, z1, ..., zn−1kX
= 1
hp
X
1≤i≤kp
kxi−L, z1, ..., zn−1kX− 1 hp
X
1≤i≤kp−1
kxi−L, z1, ..., zn−1kX
= kp
hp
1 kp
X
1≤i≤kp
kxi−L, z1, ..., zn−1kX
−kp−1
hp
1 kp−1
X
1≤i≤kp−1
kxi−L, z1, ..., zn−1kX
. (6) Now we have khp
p ≤1+νν and kp−1h
p ≤ 1ν,since hp=kp−kp−1.Hence using (6), we have x∈Nθ(X).
PROPOSITION 12. Letθ = (kp) be a lacunary sequence with lim suppηp < ∞, then Nθ(X)⊆ |σ1|(X).
PROOF. Let lim supηp<∞. Then there exists aM >0 such thatηp< M for all p≥1. Letx∈Nθ0(X) andε >0. Then we can findR >0 andK >0 such that
sup
i≥R
Si = sup
i≥R
1 hi
ki
X
i=1
kxi, z1,· · ·, zn−1kX− 1 hi
ki−1
X
i=1
kxi, z1,· · ·, zn−1kX
< ε
and Si < K for all i = 1,2, ... . Then if t is any integer withkp−1 < t ≤ kp, where p > R, we can write
1 t
t
X
i=1
kxi, z1,· · ·, zn−1kX
≤ 1
kp−1 kp
X
i=1
kxi, z1,· · ·, zn−1kX
= 1
kp−1
X
I1
kxi, z1,· · ·, zn−1kX+· · ·+X
Ip
kxi, z1,· · ·, zn−1kX
= k1
kp−1
S1+k2−k1
kp−1
S2+· · ·+kR−kR−1
kp−1
SR+kR+1−kR
kp−1
SR+1
+· · ·+kp−kp−1
kp−1
Sp
≤
sup
i≥1
Si
kR
kp−1 +
sup
i≥R
Si
kp−kR
kp−1
≤ KkkR
p−1 +εM.
Since kp−1 −→ ∞ as i −→ ∞, it follows that x ∈ |σ1|0(X). The general inclusion Nθ(X)⊆ |σ1|(X) follows by linearity.
PROPOSITION 13. Let θ = (kp) be a lacunary sequence with 1 < lim infηp ≤ lim supηp<∞, then|σ1|(X) =Nθ(X).
PROOF. Proof follows by combining Proposition 11 and Proposition 12.
3 Examples and Remarks
The concept of 2-normed spaces was introduced and studied by Siegfried G¨ahler, a German Mathematician who worked at German Academy of Science, Berlin, in a series of paper in German language published in Mathematische Nachrichten in the mid of 1960’s. Later on it was further generalized, and the notion of n-norm was introduced by Misiak. Very often G¨ahler has raised the following questions: What is the real motivation for studying 2-norm structures? Is there a physical situation or an abstract concept where norm topology does not work but 2-norm topology does? After the investigations of this paper, we can comment that while studying n-normed structure or summability methods for sequences with a realn-normed linear space as base space the main issue should be the use of the n-norms. We also observe that if a term in the definition of n-norm represents the change of shape, and the n-norm stands for the associated area or center of gravity of the term, we can think of some plausible applicable of the notion ofn-norm. As an example, we can think of use of the notion of n-norm for a process where for a particular output we needn-inputs but with one main input and other (n−1)-inputs as dummy inputs to complete the process. Keeping all these factors in mind we provide some further examples.
EXAMPLE 1. Consider the linear space Pm of real polynomials of degree≤mon the interval [0,1]. Let {xi}nmi=0 benm+ 1 arbitrary but distinct fixed points in [0,1].
Forf1, f2, . . . , fn inPm, let us define kf1, f2, . . . , fnk=
0 if f1, . . . , fn are lin. independent,
nm
P
i=0|f1(xi)f2(xi). . . fn(xi)| if f1, . . . , fn are lin. dependent.
Thenk., . . . , .kis ann-norm onPm.
PROOF. We prove only the propertykf1, f2, . . . , fnk= 0 if and only iff1, f2, . . . , fn
are linearly dependent. Other properties ofn-norm can be easily verified. Iff1, f2, . . . , fn
are linearly dependent, thenkf1, f2, . . . , fnk= 0. Conversely assume
nm
X
i=0
|f1(xi)f2(xi). . . fn(xi)|= 0.
This implies that
f1(xi)f2(xi). . . fn(xi) = 0 atnm+ 1 distinct points.
Since the degree of eachfi≤m, we must have at least onefi= 0. Thus kf1, f2, . . . , fnk= 0 if and only iff1, f2, . . . , fn are linearly dependent.
EXAMPLE 2. Consider the spaceC0 of real sequences with only finite number of non-zero terms. Let us define:
kx1, x2, ..., xnk=
0 if x1, x2, ..., xnlniearly dependent,
P∞ k=1(|xk1
||xk2
|. . .|xkn|) if x1, x2, ..., xnlinearly independent.
Then k., . . . , .k is an n-norm onC0. But it is not ann-norm on c0 andl∞ consisting of real sequences.
In view of Lemma 1, Lemma 2 and definitions of convergence and Cauchy sequence in n-norm, the concept of derived norm has special role through the subject.
Associated to the derived norm k., ..., .k∞, we can define the balls (open) B(x, ε) centered atxhaving radius εby
B(x, ε) :={y:kx−y, z2, ..., zn−1k∞< ε}, where
kx−y, z2, ..., zn−1k∞:= max{kx−y, z2, ..., zn−1, ujk:j= 1,2, ..., d}.
We may want to view ann-norm on a real linear space M, say as a norm on the Cartesian product space Mn which is invariant under permutation. But this is not true. One may find it interesting to see the differences between these two concepts through the condition (N1) in the definition of n-norm. We now give the following example which seems to be a 2-norm but not true.
EXAMPLE 3. Let Y be the space of all bounded real-valued functions onR. For f, g∈Y, let us define
kf, gk=
0 if f, g are linearly dependent, supt∈R|f(t)g(t)| if f, gare linearly independent.
Thenk., .kis not a 2-norm. To see this,
f(t) =
0 if t≤0, sint if 0< t < π,
0 if t≥π, and
g(t) =
0 if t≤ −π, sint if −π < t <0,
0 if t≥0.
Thenf andg are linearly independent. But kf, gk= 0.
EXAMPLE 4. LetX be a 2-normed space of all bounded real-valued functions on R andk.k∞be a derived norm on X. LetT :X −→X be defined by
h(t) =T f(t) =f(t−∆),
where ∆>0 is a constant. This is a model of a delay line, which is an electric device whose output h is a delayed version of the input f, the time delay be ∆. Then T is linear and bounded with respect to the derived norm.
References
[1] D. Borwein, Linear functionals connected with strong Ces`aro summability, J. Lon- don Math. Soc., 40(1965), 628–634.
[2] A. R. Freedman, J. J. Sember and M. Raphael, Some Ces`aro-type summability spaces, Proc. Lond. Math. Soc., 37(1978), 508–520.
[3] S. G¨ahler, 2-metrische R¨aume und ihre topologische Struktur, Math. Nachr., 28(1963), 115–148.
[4] S. G¨ahler, Linear 2-normietre R¨aume, Math. Nachr., 28(1965), 1–43.
[5] S. G¨ahler, Uber der Uniformisierbarkeit 2-metrische R¨aume, Math. Nachr., 28(1965), 235–244.
[6] H. Gunawan, On n-inner product,n-norms, and the Cauchy-Schwarz Inequality, Scientiae Mathematicae Japonicae (Online), 5(2001), 47–54.
[7] H. Gunawan, The space of p-summable sequences and its natural n-norm, Bull.
Aust. Math. Soc., 64(2001), 137–147.
[8] H. Gunawan and M. Mashadi, On finite Dimensional 2-normed spaces, Soochow J. of Math., 27(2001), 631–639.
[9] H. Gunawan and M. Mashadi, Onn-normed spaces, Int. J. Math. and Math. Sci., 27(2001), 631–639.
[10] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley and Sons, 1978.
[11] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford, 18(1967), 345–355.
[12] A. Misiak,n-inner product spaces, Math. Nachr., 140(1989), 299–319.
[13] A. White, 2-Banach spaces, Math. Nachr. 42(1969), 43–60.
