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ON n-NORMED SPACES
HENDRA GUNAWAN and M. MASHADI
(Received 6 August 2000 and in revised form 12 October 2000)
Abstract.Given ann-normed space withn≥2, we offer a simple way to derive an(n−1)- norm from then-norm and realize that anyn-normed space is an(n−1)-normed space.
We also show that, in certain cases, the(n−1)-norm can be derived from then-norm in such a way that the convergence and completeness in then-norm is equivalent to those in the derived (n−1)-norm. Using this fact, we prove a fixed point theorem for some n-Banach spaces.
2000 Mathematics Subject Classification. 46B20, 46B99, 46A19, 46A99, 47H10.
1. Introduction. Letn∈NandXbe a real vector space of dimensiond≥n. (Here we allowdto be infinite.) A real-valued function·, . . . ,·onXnsatisfying the follow- ing four properties
(1) x1, . . . , xn =0 if and only ifx1, . . . , xnare linearly dependent;
(2) x1, . . . , xnis invariant under permutation;
(3) x1, . . . , xn−1, αxn = |α|x1, . . . , xn−1, xnfor anyα∈R;
(4) x1, . . . , xn−1, y+z ≤ x1, . . . , xn−1, y+x1, . . . , xn−1, z,
is called ann-normonXand the pair(X,·, . . . ,·)is called ann-normed space.
A trivial example of ann-normed space is X=Rn equipped with the following n-norm:
x1, . . . , xnE:=abs
x11 ··· x1n
... . .. ... xn1 . . . xnn
, (1.1)
wherexi=(xi1, . . . , xin)∈Rnfor eachi=1, . . . , n. (The subscriptEis for Euclidean.) Note that in ann-normed space(X,·, . . . ,·), we have, for instance,x1, . . . , xn ≥0 andx1, . . . , xn−1, xn = x1, . . . , xn−1, xn+α1x1+···+αn−1xn−1for allx1, . . . , xn∈ Xandα1, . . . , αn−1∈R.
The theory of 2-normed spaces was first developed by Gähler [3] in the mid 1960’s, while that ofn-normed spaces can be found in [11]. Recent results can be found, for example, in [9,10]. Related works onn-metric spaces andn-inner product spaces may be found, for example, in [1,2,4,5,7,6,12].
In this note, we will show that everyn-normed space withn≥2 is an(n−1)-normed space and hence, by induction, an (n−r )-normed space for allr =1, . . . , n−1. In particular, given ann-normed space, we offer a simple way to derive an(n−1)-norm from then-norm, different from that in [5].
We will also apply our result to study convergence and completeness inn-normed spaces, which will be defined later. This enables us to prove a fixed point theorem for somen-normed spaces.
The casen=2 was previously studied in [8].
2. Preliminary results. Suppose hereafter that n≥2 and(X,·, . . . ,·)is an n- normed space of dimensiond≥n. Take a linearly independent set{a1, . . . , an}inX.
With respect to{a1, . . . , an}, define the following function·, . . . ,·∞onXn−1by x1, . . . , xn−1∞:=max x1, . . . , xn−1, ai:i=1, . . . , n
. (2.1)
Then we have the following result.
Theorem2.1. The function·, . . . ,·∞defines an(n−1)-norm onX.
Proof. We will verify that·, . . . ,·∞ satisfies the four properties of an(n−1)- norm.
(1) Ifx1, . . . , xn−1are linearly dependent, thenx1, . . . , xn−1 =0 for eachi=1, . . . , n, and hencex1, . . . , xn−1∞=0. Conversely, ifx1, . . . , xn−1∞=0, thenx1, . . . , xn−1, ai
=0 and accordinglyx1, . . . , xn−1,aiare linearly dependent for eachi=1, . . . , n. But this can only happen whenx1, . . . , xn−1are linearly dependent.
(2) Sincex1, . . . , xn−1, aiis invariant under any permutation of{x1, . . . , xn−1}, we find thatx1, . . . , xn−1∞is also invariant under any permutation.
(3) Observe that
x1, . . . , xn−2, αxn−1∞=max x1, . . . , xn−2, αxn−1, ai:i=1, . . . , n
= |α|max x1, . . . , xn−2, xn−1, ai:i=1, . . . , n
= |α|x1, . . . , xn−2, xn−1
∞.
(2.2)
(4) Observe that
x1, . . . , xn−2, y+z∞=max x1, . . . , xn−2, y+z, ai:i=1, . . . , n
≤max x1, . . . , xn−2, y, ai:i=1, . . . , n +max{x1, . . . , xn−2, z, ai:i=1, . . . , n
=x1, . . . , xn−2, y∞+x1, . . . , xn−2, z∞.
(2.3)
Therefore·, . . . ,·∞defines an(n−1)-norm onX.
Corollary 2.2. Every n-normed space is an (n−r )-normed space for allr = 1, . . . , n−1. In particular, everyn-normed space is a normed space.
Remark 2.3. Note that in general the function x1, . . . , xn−1p := {n
i=1x1, . . . , xn−1, aip}1/p, where 1≤p≤ ∞, also defines an(n−1)-norm onX. These(n−1)- norms, however, are equivalent to ·, . . . ,·∞, as long as we use the same set ofn vectorsa1, . . . , an. In certain cases, it is possible to get equivalent(n−1)-norms even if we use different sets ofnvectors.
2.1. The standard case. Take a look at a standard example. LetXbe a real inner product space of dimensiond≥n. EquipXwith the standardn-norm
x1, . . . , xn
S:=
x1, x1
···
x1, xn
... . .. ... xn, x1
···
xn, xn
1/2
, (2.4)
where·,·denotes the inner product onX. (IfX=Rn, then thisn-norm is exactly the same as the Euclideann-norm·, . . . ,·Ementioned earlier.)
Notice that forn=1, the aboven-norm is the usual normx1S= x1, x11/2, which gives the length ofx1, while forn=2, it defines the standard 2-norm x1, x2S = {x12Sx22S−x1, x22}1/2, which represents the area of the parallelogram spanned by x1 and x2. Further, if X =R3, then x1, x2, x3s = x1, x2, x3E is nothing but the volume of the parallelograms spanned byx1, x2, andx3. In general,x1, . . . , xnS
represents the volume of then-dimensional parallelepiped spanned byx1, . . . , xninX.
Now let{e1, . . . , en}be an orthonormal set inX. Then, byTheorem 2.1, the following function
x1, . . . , xn−1∞:=max x1, . . . , xn−1, eiS:i=1, . . . , n
(2.5) defines an(n−1)-norm onX. Further, we have the following fact.
Fact2.4. On a standardn-normed spaceX, the derived(n−1)-norm·, . . . ,·∞, defined with respect to{e1, . . . , en}, is equivalent to the standard(n−1)-norm·, . . . ,·S. Precisely, we have
x1, . . . , xn−1∞≤x1, . . . , xn−1S≤√
nx1, . . . , xn−1∞ (2.6) for allx1, . . . , xn−1∈X.
Proof. Assume thatx1, . . . , xn−1 are linearly independent. For eachi=1, . . . , n, writeei=ei◦+e⊥i whereei◦∈span{x1, . . . , xn−1}andei⊥⊥span{x1, . . . , xn−1}. Then we have
x1, . . . , xn−1, ei
S=x1, . . . , xn−1, e⊥i
S
=
x1, x1 ··· x1, xn−1 0
... . .. ... ...
xn−1, x1 ··· xn−1, xn−1 0
0 ··· 0
e⊥i, e⊥i
1/2
≤
x1, x1 ··· x1, xn−1 ... . .. ... xn−1, x1 ··· xn−1, xn−1
1/2
=x1, . . . , xn−1S.
(2.7)
Hence we getx1, . . . , xn−1∞≤ x1, . . . , xn−1S.
Next, take a unit vectore=α1e1+···+αnensuch thate⊥span{x1, . . . , xn−1}. (Here we are still assuming thatx1, . . . , xn−1are linearly independent.) Then, by properties (3) and (4) of then-norm, we have
x1, . . . , xn−1S=x1, . . . , xn−1, eS
≤α1x1, . . . , xn−1, e1S+···+αnx1, . . . , xn−1, enS
≤α1+···+αnx1, . . . , xn−1∞.
(2.8)
But, by the Cauchy-Schwarz inequality, we have n
i=1
αi≤
n i=1
12
1/2
n i=1
αi2
1/2
=√
n. (2.9)
Hence we obtain
x1, . . . , xn−1S≤√
nx1, . . . , xn−1∞, (2.10)
and this completes the proof.
2.2. The finite-dimensional case. For finite-dimensional n-normed space (X,
·, . . . ,·), we can in general derive an(n−1)-norm from then-norm in the following way. Take a linearly independent set{a1, . . . , am}inX, withn≤m≤d. With respect to{a1, . . . , am}, define the following function·, . . . ,·∞onXn−1by
x1, . . . , xn−1∞:=max x1, . . . , xn−1, ai:i=1, . . . , m
. (2.11)
Then, as inTheorem 2.1, the function·, . . . ,·∞defines an(n−1)-norm onX.
As we will see later, we can obtain a better(n−1)-norm by using a set ofd, rather than justn, linearly independent vectors inX(that is, by using a basis forX).
3. Applications and further results. Recall that a sequencex(k)in ann-normed space(X,·, . . . ,·)is said toconvergeto anx∈X(in then-norm) whenever
k→∞limx1, . . . , xn−1, x(k)−x=0 (3.1) for everyx1, . . . , xn−1∈X.
The following proposition says that the convergence in then-norm implies the con- vergence in the derived(n−1)-norm·, . . . ,·∞, defined with respect to an arbitrary linearly independent set{a1, . . . , an}inX.
Proposition 3.1. If x(k)converges to an x∈X in the n-norm, then x(k)also converges toxin the derived(n−1)-norm·, . . . ,·∞, that is,
k→∞limx1, . . . , xn−2, x(k)−x∞=0 (3.2) for everyx1, . . . , xn−2∈X.
Proof. Ifx(k)converges to anx∈Xin then-norm, then
k→∞limx1, . . . , xn−2, x(k)−x, ai=0 (3.3)
for everyx1, . . . , xn−2∈Xandi=1, . . . , n, and hence
k→∞limx1, . . . , xn−2, x(k)−x
∞=0 (3.4)
for everyx1, . . . , xn−2∈X, that is,x(k)converges tox in the derived(n−1)-norm
·, . . . ,·∞.
3.1. The standard case. In a standardn-normed space(X,·, . . . ,·S), the converse ofProposition 3.1is also true, especially when the derived(n−1)-norm·, . . . ,·∞is defined with respect to an orthonormal set{e1, . . . , en}inXas inSection 2.1.
Fact3.2. A sequence in a standardn-normed spaceXis convergent in then-norm if and only if it is convergent in the derived(n−1)-norm·, . . . ,·∞.
Proof. Suppose that x(k)converges to an x∈ X in the derived (n−1)-norm
·, . . . ,·∞. We want to show that x(k) also converges to x in the n-norm. Take x1, . . . , xn−1∈X. Then one may observe that
x1, . . . , xn−2, xn−1, x(k)−xS≤x1, . . . , xn−2, x(k)−xSxn−1S, (3.5) where·, . . . ,·S and · S on the right-hand side denote the standard(n−1)-norm and the usual norm onX, respectively. ByFact 2.4, we have
x1, . . . , xn−2, xn−1, x(k)−xS≤√
nx1, . . . , xn−2, x(k)−x
∞xn−1S. (3.6) But limk→∞x1, . . . , xn−2, x(k)−x∞=0, and so we conclude that
k→∞limx1, . . . , xn−1, x(k)−xS=0, (3.7) that is,x(k)converges toxin then-norm.
Corollary3.3. A sequence in a standardn-normed space is convergent in then- norm if and only if it is convergent in the standard(n−1)-norm and, by induction, in the standard(n−r )-norm for allr=1, . . . , n−1. In particular, a sequence in a standard n-normed space is convergent in then-norm if and only if it is convergent in the usual norm·S:= ·,·1/2.
3.2. The finite-dimensional case. We also have a similar result for finite- dimensional n-normed space (X,·, . . . ,·). Let {b1, . . . , bd} be a basis for X. With respect to{b1, . . . , bd}, define the following function·, . . . ,·onXn−1by
x1, . . . , xn−1:=max x1, . . . , xn−1, bi:i=1, . . . , d
. (3.8)
Then, as mentioned before, the function·, . . . ,·defines an(n−1)-norm onX.
With this derived(n−1)-norm, we have the following result.
Proposition3.4. A sequence in the finite-dimensionaln-normed spaceXis conver- gent in then-norm if and only if it is convergent in the derived(n−1)-norm·, . . . ,·.
Proof. If a sequence inX is convergent in the n-norm, then it will certainly be convergent in the(n−1)-norm·, . . . ,·. Conversely, suppose thatx(k)converges to anx∈Xin ·, . . . ,·. Takex1, . . . , xn−1∈X. Writingxn−1=α1b1+ ··· +αdbd, we get
x1, . . . , xn−2, xn−1, x(k)−x≤α1x1, . . . , xn−2, x(k)−x, b1 +···+αdx1, . . . , xn−2, x(k)−x, bd
≤α1+···+αdx1, . . . , xn−2, x(k)−x. (3.9)
But limk→∞x1, . . . , xn−2, x(k)−x=0, and so we obtain
k→∞limx1, . . . , xn−1, x(k)−x=0, (3.10) that is,x(k)converges toxin then-norm.
3.3. The standard, separable case. We go back to the standard case, whereX is a real inner product space of dimensiond≥nequipped with the standardn-norm
·, . . . ,·S as inSection 2.1. But suppose now thatXis separable and that{ei:i∈Id}, whereId:= {1, . . . , d}(if d <∞) orN(ifd= ∞), is an orthonormal basis forX. For everyx1, . . . , xn−1∈Xand every basis vectorei(i∈Id), we have
x1, . . . , xn−1, eiS≤x1, . . . , xn−1S, (3.11) where ·, . . . ,·S on the right-hand side denotes the standard (n−1)-norm on X.
Hence, with respect to {ei:i∈Id}, we may define the function·, . . . ,· onXn−1 by x1, . . . , xn−1:=sup x1, . . . , xn−1, eiS:i∈Id
(3.12)
and check that it also defines an(n−1)-norm onX. Moreover, we have the following relation between the two derived(n−1)-norms·, . . . ,·and ·, . . . ,·∞(the latter being defined with respect to{e1, . . . , en}only):
x1, . . . , xn−1∞≤x1, . . . , xn−1≤x1, . . . , xn−1S≤√
nx1, . . . , xn−1∞ (3.13) for everyx1, . . . , xn−1∈X. Hence we conclude the following fact.
Fact3.5. On a standardn-normed spaceX, the two derived(n−1)-norms·, . . . ,·∞
and·, . . . ,·and the standard(n−1)-norm·, . . . ,·S are equivalent. Accordingly, a sequence in a standardn-normed spaceXis convergent in then-norm if and only if it is convergent in one of the three(n−1)-norms.
3.4. Cauchy sequences, completeness and fixed point theorem. Recall that a se- quencex(k)in ann-normed space(X,·, . . . ,·)is calledCauchy(with respect to the n-norm) if
k,l→∞lim x1, . . . , xn−1, x(k)−x(l)=0 (3.14)
for everyx1, . . . , xn−1∈X. If every Cauchy sequence inXconverges to anx∈X, then Xis said to becomplete(with respect to then-norm). A completen-normed space is then called ann-Banach space.
By replacing the phrases “x(k)converges tox” with “x(k)is Cauchy” and “x(k)−x”
with “x(k)−x(l),” we see that the analogues ofProposition 3.1,Fact 3.2,Corollary 3.3, Proposition 3.4, andFact 3.5hold for Cauchy sequences.
Hence, for the standard or finite-dimensional case, we have the following result.
Proposition3.6. (a)A standardn-normed space is complete if and only if it is com- plete with respect to one of the three(n−1)-norms·, . . . ,·∞,·, . . . ,·, or·, . . . ,·S. By induction, a standardn-normed space is complete if and only if it is complete with respect to the usual norm·S:= ·,·1/2.
(b)A finite-dimensionaln-normed space is complete if and only if it is complete with respect to the derived(n−1)-norm·, . . . ,·.
Consequently, we have the following result.
Corollary 3.7 (fixed point theorem). Let(X,·, . . . ,·) be a standard or finite- dimensional n-Banach space, and T a contractive mapping of X into itself, that is, there exists a constantC∈(0,1)such that
x1, . . . , xn−1, T y−T z≤Cx1, . . . , xn−1, y−z (3.15)
for allx1, . . . , xn−1, y, zinX. ThenT has a unique fixed point inX.
Proof. First consider the casen=2 (see [8]). ByProposition 3.6, we know thatX is a Banach space with respect to the derived norm·∞(for standard case) or·
(for finite-dimensional case). Since the mappingT is also contractive with respect to
·∞or·, we conclude by the fixed point theorem for Banach spaces thatT has a unique fixed point inX. Forn >2, the result follows by induction.
Remark3.8. In the finite-dimensional case, it is actually enough to assume thatX is ann-normed space because we know that all finite-dimensional normed spaces are complete and, byProposition 3.6(b), so are all finite-dimensionaln-normed spaces.
4. Concluding remark. We have shown that an n-normed space with n≥ 2 is an (n−1)-normed space and that, for the standard or finite-dimensional case, the (n−1)-norm can be derived from then-norm in such a way that the convergence and completeness in then-norm is equivalent to those in the derived(n−1)-norm.
Below is an example of a non-standard, infinite-dimensional 2-normed space for which we can derive a norm from the 2-norm such that the convergence and com- pleteness in the 2-norm is equivalent to those in the derived norm.
Let X=l∞, the space of bounded sequences of real numbers. EquipX with the following 2-norm
x, y:=sup
i∈Nsup
j∈N
xiyj−xjyi, (4.1) wherex=(x1, x2, x3, . . .)andy=(y1, y2, y3, . . .). Leta1=(1,0,0, . . .)anda2=(0,1,0, . . .).
With respect to{a1, a2}, we derive the norm·∞via
x∞:=max x, a1,x, a2. (4.2) Butx, a1 =supi∈N\{1}|xi|andx, a2 =supi∈N\{2}|xi|, and so we obtain
x∞=sup
i∈N
xi, (4.3)
the usual norm onl∞.
Now suppose thatx(k)is a sequence inXthat converges toxin the derived norm
·∞. For everyy∈X, we have x(k)−x, y =sup
i∈Nsup
j∈N
xi(k)−xi
yj−
xj(k)−xj
yi
≤sup
i∈Nsup
j∈N
xi(k)−xiyj+xj(k)−xjyi
≤2x(k)−x∞y∞,
(4.4)
whence limk→∞x(k)−x, y =0. Hencex(k)converges toxin the 2-norm·,·. Thus, for this particular example, we see that the convergence in the 2-norm is equivalent to that in the derived norm. By similar arguments, we can also verify that the completeness in the 2-norm is equivalent to that in the derived norm.
For general non-standard, infinite-dimensional n-normed spaces, however, it is unknown whether we can always derive an(n−1)-norm from then-norm such that the convergence and completeness in then-norm is equivalent to those in the derived (n−1)-norm.
Acknowledgements. This paper was revised during Gunawan’s visit to the School of Mathematics, UNSW, Sydney, in 2000/2001, under an Australia-Indonesia Merdeka Fellowship, funded by the Australian Government through the Department of Education, Training and Youth Affairs and promoted through Australia Education International.
Both authors would like to thank the anonymous referees for their useful comments and suggestions.
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Hendra Gunawan: Department of Mathematics, Bandung Institute of Technology, Bandung40132, Indonesia
E-mail address:[email protected]
Mashadi: Department of Mathematics, University of Riau, Pekanbaru 28293, Indonesia
