322 HENDRA GUNAWAN AND MASHADI èivè kx; y + zk çkx; yk + kx; zk: The pair èx; kæ; ækè is then called a 2-normed space. A standard example of a 2-norme

Transcription

1 SOOCHOW JOURNAL OF MATHEMATICS Volume 27, No. 3, pp , July 2001 ON FINITE DIMENSIONAL 2-NORMED SPACES BY HENDRA GUNAWAN AND MASHADI Abstract. In this note, we shall study ænite dimensional 2-normed spaces and show that their topology can be fully described by using a certain norm derived from the 2-norm. Moreover, we show that a 2-Banach space is a Banach space and use this fact to prove the Fixed-Point Theorem. Our result extends to some inænite dimensional 2-normed spaces. 1. Introduction The concepts of 2-metric spaces and 2-normed spaces were intially introduced by Gíahler ë 5, 6, 7ë in 1960's. Since then, many researchers have studied these two spaces and obtained various results, see for instance ë1, 2, 9, 10, 11, 12ë. One well-known fact about 2-normed spaces is that they are actually normed spaces. In this note, we shall study ænite dimensional 2-normed spaces and show that their topology can be fully described by using a certain norm derived from the 2-norm. Moreover, we shall show that a 2-Banach space is a Banach space and use this fact to prove results such as the Fixed-Point Theorem. We also indicate that our result extends to some inænite dimensional 2-normed spaces. Let X be a real vector space of dimension d, where 2 ç d é 1. A 2- norm on X is a function kæ; æk : X æ X! R which satisæes the following four conditions: èiè kx; yk = 0 if and only if x and y are linearly dependent; èiiè kx; yk = ky; xk; èiiiè kx;æyk = jæjkx; yk; æ 2 R; Received March 16, 2000; revised September 19, AMS Subject Classiæcation. 46B20, 46B99, 47H10, 54H25. Key words. 2-normed spaces, normed spaces, æxed-point theorem. 321

2 322 HENDRA GUNAWAN AND MASHADI èivè kx; y + zk çkx; yk + kx; zk: The pair èx; kæ; ækè is then called a 2-normed space. A standard example of a 2-normed space is R 2 equipped with the following 2-norm kx; yk := the area of the triangle having vertices 0; x; and y: Observe that in any 2-normed space èx; kæ; ækè we have kx; yk ç 0 and kx; y + æxk = kx; yk for all x; y 2 X and æ 2 R. Also, if x; y and z are linearly dependent èthis happens, for instance, when d = 2è, then kx; y + zk = kx; yk + kx; zk or kx; y, zk = kx; yk + kx; zk. Given a 2-normed space èx; kæ; ækè, one can derive a topology for it via the following deænition of the limit of a sequence: A sequence èx n èinx is said to be convergent to x in X if lim kx n, x; yk = 0 for every y 2 X. In such a case, we write lim x n := x and call x the limit of èx n è. The uniqueness of the limit of a convergent sequence can be veriæed as follows. Suppose èx n è is convergent to two distinct limits x and y in X. Choose z 2 X such that kx,y; zk 6= 0 and take N 2 N suæciently large such that kx N,x; zk é 1 2 kx, y; zk and kx N, y; zk é 2 1 kx, y; zk simultaneously. Then, by the triangle inequality, we obtain kx, y; zk çkx, x N ; zk + kx N, y; zk é 1 2 kx, y; zk + 1 kx, y; zk = kx, y; zk; 2 which is absurd. Hence, whenever it exists, lim x n must be unique. Now that we have deæned convergent sequences in a 2-normed space, it is natural to inquire how we can deæne balls there so that we can fully describe its topology by using these balls. In the next section, we show how to do this for ænite dimensional case. Moreover, we show that a 2-Banach space is a Banach space and use this fact to prove the Fixed-Point Theorem. In the last section we indicate that our result extends to some inænite dimensional 2-normed spaces. 2. Main Results Suppose hereafter that èx; kæ; ækè is a 2-normed space. Recall that we assume X to have dimension d, where 2 ç d é 1, unless otherwise stated. Fix fu 1 ;:::;u d g to be a basis for X. Then we have the following:

3 ON FINITE DIMENSIONAL 2-NORMED SPACES 323 Lemma 2.1. A sequence èx n è in X is convergent to x in X if and only if lim kx n, x; u i k =0for every i =1;:::;d. Proof. It suæces to prove that if lim kx n,x; u i k = 0 for every i =1;:::;d, then we have lim kx n, x; yk = 0 for every y 2 X. But this is clear since every y 2 X can be written as y = æ 1 u 1 + :::+ æ d u d for some æ 1 ;:::;æ d 2 R, and by the triangle inequality we have for all n 2 N. kx n, x; yk çjæ 1 jkx n, x; u 1 k + æææ+ jæ d jkx n, x; u d k Following Lemma 2.1, we have: Lemma 2.2. A sequence èx n è in X is convergent to x in X if and only if lim maxfkx n, x; u i k : i =1;:::;dg =0. This simple fact enlightens us to deæne a norm on X as follows: With respect to the basis fu 1 ;:::;u d g, we can deæne a norm on X, which we shall denote it by kæk 1,by kxk 1 := maxfkx; u i k : i =1;:::;dg: Indeed, one may observe that: èiè kxk 1 =0if and only if x = 0, èiiè kæxk 1 = jæjkxk 1, and èiiiè kx + yk 1 çkxk 1 + kyk 1 for all x; y 2 X and æ 2 R. èone might like to compare our norm to the one oæered by Gíahler in ë5ë or ë6ë.è by Note that for 1 ç p ç1in general, we can deæne the function kæk p on X kxk p := n dx kx; u i k po 1=p ; and check that it also deænes a norm on X. But, since X is ænite dimensional, all these norms are equivalent, and so throughout this section we shall only work with kæk 1, unless otherwise required. Note also that the choice of the basis here is not essential. If we choose another basis for X, say fv 1 ;:::;v d g, and deæne the norm kæk 1 with respect to it, then the resulting norm will be equivalent to the one deæned with respect to fu 1 ;:::;u d g.

4 324 HENDRA GUNAWAN AND MASHADI Using the derived norm kæk 1, Lemma 2.2 now reads: Lemma 2.2 y. A sequence èx n è in X is convergent to x in X if and only if lim kx n, xk 1 =0. Associated to the derived norm kæk 1,we can deæne the èopenè balls B fu1 ;:::;u d g èx, rè centered at x having radius r by Using these balls, Lemma 2.2 y becomes: B fu1 ;:::;u d gèx;rè:=fy : kx, yk 1 érg: Lemma 2.2 z. A sequence èx n è is convergent to x in X if and only if 8 æé 0 9 N 2 N such that n ç N è x n 2 B fu1 ;:::;u d gèx;æè. Summarizing all these results, we have: Theorem 2.3. Any ænite dimensional 2-normed space is a normed space and its topology agrees with that generated by the derived norm kæk 1. As we shall see below, many results in a normed space can be veriæed analogously in a 2-normed space by using the derived norm kæk 1 andèor its associated balls. But ærst let us see some examples. Example 2.4. Let X = R 2 be equipped with the 2-norm kx; yk := the area of the parallelogram spanned by the vectors x and y è= twice the area of the triangle having vertices 0, x and yè, whichmay be given explicitly by the formula kx; yk = jx 1 y 2, x 2 y 1 j; x =èx 1 ;x 2 è; y =èy 1 ;y 2 è: Take the standard basis fi; jg for R 2. Then, kx; ik = jx 2 j and kx; jk = jx 1 j, and so the derived norm kæk 1 with respect to fi; jg is kxk 1 = maxfjx 1 j; jx 2 jg; x =èx 1 ;x 2 è: Thus the derived norm kæk 1 here is exactly the same as the uniform norm on R 2. Accordingly, the ball B fi;jg èx;rè is a square centered at x having ëradius" r. Since the derived norm is equivalent to the Euclidean norm on R 2,we conclude that R 2 equipped with the above 2-norm is nothing but the Euclidean plane.

5 ON FINITE DIMENSIONAL 2-NORMED SPACES 325 More generally, we have: Fact 2.5. The 2-normed space èr d ; kæ; ækè where kx; yk := the area of the parallelogram spanned by x and y is a normed space whose norm is equivalent to the Euclidean's. Proof. For every x = èx 1 ;:::;x d è and y = èy 1 ;:::;y d è 2 R d, the 2-norm kx; yk may be given explicitly by the formula kx; yk = çç dx x 2 i çç dx y 2 j j=1 ç, ç dx ç 2 ç 1=2 x i y i : Let fe 1 ;:::;e d g be the standard basis for R d. Then, for each j = 1;:::;d, we have kx; e j k = nç dx x 2 i ç, x 2 jo 1=2; and so the derived norm kæk 1 with respect to fe 1 ;:::;e d g is kxk 1 = max çnç dx x 2 i ç o, x 2 1=2 ç j : j =1;:::;d : Now let kæk E denote the Euclidean norm on R d. Then it is easy to check that kxk 1 çkxk E ç p 2kxk 1 for all x 2 R d, conærming that the derived norm is equivalent to the Euclidean's. Remark. For the above 2-normed space èr d ; kæ; ækè, it is worth observing P that kxk 2 = d P kx; e j k 2 = èd, 1è d x 2 i, that is, kæk 2 is a multiple of the j=1 Euclidean norm. We are now about to prove the Fixed Point Theorem for ænite dimensional 2-Banach spaces. èrecall that èx; kæ; ækè is a 2-Banach space if every Cauchy sequence in X, that is any sequence èx n èinx such that lim kx m,x n ; yk =0 m; for every y 2 X, is convergent to some x in X.è But ærst we need the following: Lemma 2.6. èx; kæ; ækè is a 2-Banach space if and only if èx; kæk 1 è is a Banach space.

6 326 HENDRA GUNAWAN AND MASHADI Proof. Since, by Lemma 2.2, the convergence in the 2-norm is equivalent to that in the derived norm, it suæces to show that èx n è is Cauchy with respect to the 2-norm if and only if it is Cauchy with respect to the derived norm. But this is clear since èx n è is Cauchy with respect to the 2-norm if and only if lim kx m, x n ; yk = 0 for every y 2 X, if and only if m; for every i =1;:::;d, if and only if is Cauchy with respect to the derived norm. Following Lemma 2.6, we have: lim kx m, x n ; u i k =0 m; lim kx m, x n k 1 =0,if and only if èx n è m; Corollary 2.7.èFixed Point Theoremè Suppose èx; kæ; ækè is a 2-Banach space. Let T be a self-mapping of X such that kt x, T y; zk çkkx, y; zk for all x; y; z in X, where k is a constant in è0; 1è. point in X. Then T has a unique æxed Proof. With respect to the derived norm kæk 1, the mapping T satisæes kt x, T yk 1 ç kkx, yk 1 for all x; y in X. Since èx; kæk 1 è is also a Banach space, we conclude by the Fixed Point Theorem for Banach spaces that T has a unique æxed point in X. 3. Further Results We shall now show that our result extends to any separable inner-product space X èwhichmay be of inænite dimensionè equipped with the standard 2-norm kx; yk := n kxk 2 kyk 2,hx; yi 2o 1=2 ; where hæ; æi denotes the inner-product and kxk := hx; xi 1=2 is its induced norm on X èsee ë8ë for discussion on the triangle inequality for this 2-normè. First we verify an analogue of Fact 2.5. Let èe i è, indexed by a countable set I ç f1; 2g, be an orthonormal basis for X. Then, for each i 2 I, we have

7 kx; e i k = ON FINITE DIMENSIONAL 2-NORMED SPACES 327 n kxk 2,hx; e i i 2o 1=2 ç kxk. Hence we may deæne the derived norm kæk 1 with respect to èe i èby kxk 1 := sup fkx; e i k : i 2 Ig: Clearly kxk 1 çkxk for all x 2 X. Conversely, using Bessel's inequality, wehave kxk 2 çkxk 2,hx; e 1 i 2 + kxk 2,hx; e 2 i 2 = kx; e 1 k 2 + kx; e 2 k 2 ç 2kxk 2 1 ; and hence kxk ç p 2kxk 1 for all x 2 X. This shows that kæk 1 is equivalent to the already existing norm kækon X. Further, we see that the analogue of Lemma 2.1 is still valid. That is, a sequence èx n èinx is convergent tox in X if and only if lim kx n,x; e i k = 0 for every i 2 I. Indeed, given lim kx n,x; e i k = 0 for every i 2 I, we can show that lim kx n, x; yk = 0 for every y 2 X. Observe that kx n, x; yk çkx n, xkkyk for every y 2 X. Again, by Bessel's inequality, wehave kx n, xk 2 çkx n, x; e 1 k 2 + kx n, x; e 2 k 2 for all n 2 N. Hence lim kx n, xk =0,and therefore lim kx n, x; yk =0for every y 2 X. As it turns out, we obtain the following result: Fact 3.1. A sequence èx n è in X is convergent to x in X if and only if lim kx n, x; e i k =0for i =1and 2 only. X by Accordingly, with respect to fe 1 ; e 2 g,we can deæne a simpler norm kæk 2 on kxk 2 := fkx; e 1 k 2 + kx; e 2 k 2 g 1=2 : This is not surprising since we can in general use two linearly dependent vectors to deæne a norm on any 2-normed space. What is remarkable here is that the topology generated by kæk 2 agrees with that generated by the 2-norm. Observe that kxk 1 çkxk çkxk 2 ç p 2 kxk 1 for all x 2 X. This conærms that kæk 2 and kæk 1 are equivalent, and both of them are equivalent to the already existing norm kæk on X. Hence the analogue

8 328 HENDRA GUNAWAN AND MASHADI of Lemma 2.2 èand its variantsè, Lemma 2.6 and Corollary 2.7 are still valid for X, whether it is equipped with kæk 1, kæk 2 or kæk. Concluding Remark. One might realize that the 2-norm discussed in the preceding section satisæes the parallelogram identity kx + y; zk 2 + kx, y; zk 2 =2èkx; zk 2 + ky; zk 2 è: Another remarkable fact about the derived 2-norm kæk 2 is that it inherits the identity kx + yk kx, yk 2 2 =2èkxk kyk 2 2è: This observation tells us that we can also derive an inner product from a 2-inner product èsee ë3ë and ë4ë for the concept of 2-inner product spacesè and conclude that any 2-inner product space is actually an inner product space. Further results in this direction will appear elsewhere. Acknowledgment The authors would like to thank the anonymous referees for their useful comments and suggestions. References ë1ë Y. J. Cho, Fixed points for compatible mappings of type èaè, Math. Japon., 38:3è1993è, ë2ë Y. J. Cho, N. J. Huang and X. Long, Some æxed point theorems for nonlinear mappings in 2-Banach spaces, Far East J. Math. Sci., 3:2è1995è, ë3ë C. Diminnie, S. Gíahler and A. White, 2-inner product spaces, Demonstratio Math., 6è1973è, ë4ë C. Diminnie, S. Gíahler and A. White, 2-inner product spaces. II, Demonstratio Math., 10 è1977è, ë5ë S. Gíahler, 2-metrische Ríaume und ihre topologische Struktur, Math. Nachr., 26è1963è, ë6ë S. Gíahler, Lineare 2-normietre Ríaume, Math. Nachr., 28è1965è, ë7ë S. Gíahler, Uber der Uniformisierbarkeit 2-metrische Ríaume, Math. Nachr., 28è1965è, ë8ë H. Gunawan and W. Setya-Budhi, On the triangle inequality for the standard 2-norm, to appear in Majalah Ilmiah Himpunan Matematika Indonesia. ë9ë K. Iseki, Fixed point theorems in 2-metric spaces, Math. Seminar Notes, Kobe Univ., 3è1975è,

9 ON FINITE DIMENSIONAL 2-NORMED SPACES 329 ë10ë K. Iseki, Fixed point theorems in Banach spaces, Math. Seminar Notes, Kobe Univ., 2è1976è, ë11ë M. S. Khan and M. D. Khan, Involutions with æxed points in 2-Banach spaces, Internat. J. Math. Math. Sci., 16è1993è, ë12ë D. T. Xiong, Y. T. Li and N. J. Huang, 2-Metric Spaces Theory, Shanxi Normal University Publishing House, Xian, P. R. China, Department of Mathematics, Bandung Institute of Technology, Bandung 40132, Indonesia. Department of Mathematics, University of Riau, Pekanbaru 28293, Indonesia.