A CHARACTERIZATION OF STRICTLY CONVEX 2-NORMED SPACE

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1 Математички Билтен ISSN X (print) Vol. 40(LXVI) No. 1 ISSN (online) 016 (63-7) UDC Скопје, Македонија A CHARACTERIZATION OF STRICTLY CONVEX -NORMED SPACE Samoil Malčeski 1, Katerina Anevska and Risto Malčeski 3 Abstract. The terms of -norm and -normed are given b S. Gähler in the paper [10], Ch. Diminnie, S. Gahler and А. White ([3]) gave the term of strictl convex -normed spaces. This paper consists of several characterizations of strictl convex -normed spaces. 1. INTRODUCTION Let L be a real vector space with dimension greater than 1 and, be a real function on L L such that: a) x, 0 if and onl if the set { x, } is linearl dependent; b) x,, x, for all x, L ; c) x, x,, for all x, L and for each R, d) x, z x, z, z, for all x,, z L. The function, is said a -norm on L, and ( L,, ) is said a vector - normed space ([10]). Some of the fundamental properties of the -norm are the following: 1. x, 0, for all x, L and. x, x x,, for all x, L and for each R. Let n 1 be a positive integer, L be a real vector space, diml n and (, ) be a real function on L L L such that i) ( x, x ) 0, for all x, L and ( x, x ) 0 if and onl if x and are linearl dependent; ii) ( x, z) (, x z), for all x,, z L, iii) ( x, x ) (, x), for all x, L ; 010 Mathematics Subject Classification. Primar: 46B0 Secondar: 46C05 Ke words and phrases. strictl convex space, end point, minimal point with respect to a set 63

2 64 S. Malčeski, K. Anevska, R. Malčeski iv) ( x, z) ( x, z), for all x,, z L and for each R ; and v) ( x x1, z) ( x, z) ( x1, z), for all x, x1,, z L. The function (, ) is said a -inner product, and ( L,(, )) is said a -pre- Hilbert space ([1]). The concepts of -norm and -inner product are two-dimensional analogous to the concepts of a norm and an inner product. R. Ehret proved ([7]), that if ( L,(, )) is a -pre-hilbert space, then 1/ x, ( x, x ) defines a -norm. Thus, we get a vector -normed space ( L,, ) and moreover, for all x,, z L it is true that a b, c a b, c 4 ( a, b c), (1) x, z x, z ( x, z, z ), () The equalit () is actuall two-dimensional analog to the parallelogram equalit and it is said parallelepiped equalit. Further, if ( L,, ) is a vector - normed space such that for all x,, z L, (1) holds true, then () defines - inner product on L, whereb for all a, b L, x, ( x, x ) 1/, holds true Let z be a fixed non-null element of L, V() z be a subspace of L generated b z and L z be a factor space L / V ( z ). Let x z be the class of equivalence of x with respect to V() z. Clearl, L z is a vector space in which the operations vector addition and scalar multiplication are defined as following xz z ( x ) z and xz ( x) z. In [6] is proven that xz z x, z defines a norm on L z. Let x, L be non-null elements and let V ( x, ) be the subspace of L generated b the vectors x and. The vector -normed space ( L,, ) is called a strictl convex if x, z, z, z 1 and z, for x,, z L, implies ([3]). Several characterizations of strictl convex - normed space are given in papers [1], [5] [8], [1] [14], [16], [17], [3] and [4], and a few of them will be stated in the following theorem. Theorem 1. Let ( L,, ) be a -normed space. The following claims are equivalent: 1) ( L,, ) is strictl convex ) For an non-null z L, the space L z is strictl convex

3 A characterization of strictl convex -normed space 65 3) If x, z x, z, z and z, for x,, z L then x for some 0. 4) If x u, z x, z, u, z (1 ) x, z, (0,1) and z V ( x u, u), then u (1 ) x. 5) if x, z, z 1, and z, for x,, z L, then, 1 z. Example 1 ([3]). Let L be -prehilber space and x,, z L be such that xz, z, 1, and z. Then the parallelepiped equalit implies But, and z, thus, 1 z, z, z 1. (3), 0 z and the equalit (3) implies. Finall, thereb Theorem 1, L is strictl convex. Example ([18]). In the set of bounded series of real numbers l x, sup ij, N i j xi i xj j, x ( xi ) i 1, ( i ) i 1 l defines a -norm. That is, ( l,, ) is a real -normed space. The vectors x (1 1,1 1,...,1 1,...), (0, n 1,1,...,1,...) and n z (1,0,0,...,0,...) satisf x, z, z, z 1 and z, but. Therefore, l is not strictl convex -normed space.. MAINS RESULTS Let x, L. The set [ x, ] { x (1 ) [0,1]} is said a line segment (segment) with end points x and. The set ( x, ) { x (1 ) (0,1)} is said an opened line segment with end points x and.

4 66 S. Malčeski, K. Anevska, R. Malčeski Corollar 1. Let ( L,, ) be a -normed space. The following claims are equivalent: 1) ( L,, ) is strictl convex ) If x, z x, z, z and z, for x,, z L, then the set [ x, ] { x (1 ) [0,1]} is linearl dependent. Proof. Let the condition ) holds true. Since x, [ x, ], the set { x, } is linearl dependent, i.e. it exists R such that x. If we substitute in the condition x, z x, z, z and consider that z, we get 1 1, which implies 0. Thus, x for some 0, so Theorem 1 implies that L is strictl convex. Let L be a strictl convex space. If x, z x, z, z and z, then Theorem 1 implies that x for some 0. Let xt tx (1 t), for t [0,1]. Thus, xt ( t (1 t) ) x, for t [0,1] and thereb all tp, [0,1] satisf t (1 t) 0 and p (1 p) 0 we get t (1 t) t (1 t) t ( (1 ) ) ( (1 ) ) p (1 p) p p (1 p) p x t t x p p x x, So, the set { x, x } is linearl dependent, for all tp, [0,1]. Thus, the set t p [ x, ] { x (1 ) [0,1]} is linearl dependent. Let L be a -normed space, x, z L and r 0. The set B ( x, r) { L x, z r} z is called an opened ball with respect to z centered at x and radius r. If x 0 and r 1, then B z (0,1) is called a unit opened ball with respect to z. The set B [ x, r] { L x, z r} z is called a closed ball with respect to z centered at x and radius r. If x 0 and r 1, then B z[0,1] is called a unit closed ball with respect to z. The set S ( x, r) { L x, z r} z is called a sphere with respect to z centered at x and radius r. If x 0 and r 1, then S z (0,1) is called a unit closed sphere with respect to z. Clearl, Bz( x, r) Bz[ x, r] and Bz[ x, r] Bz ( x, r) Sz( x, r). Lemma 1. Let L be a -normed space and x,, z L. If x, z x, z, z,

5 A characterization of strictl convex -normed space 67 then, for all ts, 0 holds true. If z, then [ x,,, ] Sz (0,1). x z z tx s, z t x, z s, z, (4) Proof. Let 0 s t. Then the properties of the -norm impl t x, z s, z tx s, z t( x ) ( t s), z t( x ), z ( t s), z t x, z ( t s), z t x, z t, z t, z s, z t x, z s, z, i.e. in this case the equalit (4) holds true. Analogousl can be considered a case for 0 t s. Let z. If [0,1], then the equalit (4) implies Therefore, x x, z, z x 1 x, z, z x, z, z (1 ), z x, z, z 1. [, ] S (0,1). z Theorem. The -normed space L is strictl convex if and onl if x, z, z 1 and [ x, ] S z (0,1) impl. Proof. Let L be a strictl convex -normed space and let the conditions x, z, z 1 and [ x, ] S z (0,1) be satisfied. Then, [ x, ] S z (0,1), for 1 implies 1 1 x (1 ) S (0,1) z. That is, 1 z. Moreover, since L is strictl convex, we get that. Conversel, let x, z, z 1 and [ x, ] S z (0,1) impl. Let s assume that x, z x, z, z and z. Thus, Lemma 1 implies x x, z, z [, ] Sz (0,1), which according to assumption means, x z, z, i.e. xz, z, space.. Finall, the Theorem 1 implies L is strictl convex -normed Theorem 3. A -normed space L is strictl convex if and onl if the following condition is satisfied x

6 68 S. Malčeski, K. Anevska, R. Malčeski x, S (0,1), x implies z x, z 1, for, 0 and 1. Proof. Let the condition (5) be satisfied and x, z, z 1, and z. Then x, S z (0,1) and since 1 we get that, 1 z, which according to theorem 1 means that L is strictl convex. Conversel, let s assume that there exist x, Sz(0,1), x and, 0, 1 such that x, z =1. The latter implies that there exist x,, z L such that x, z, z 1, and, 0, 1 such that Since Lemma 1, x, z x, z, z. t( x) s( ), z t x, z s, z, holds true for all ts, 0. For t 1, s 1, in the last equalit, we get that there exist x,, z L such that x, z, z 1, and, 1 z. The latter, according to theorem 1, means that the space L is not strictl convex. In the purpose of the next characterization of strictl convex -normed space we will use the extremal points of the convex sets. Let C be a convex set into -normed space L. The point z C is said to be an extremal (end) point for the set C if z tx (1 t), for some t (0,1) and some x, C implies. Theorem 4. - normed space L is strictl convex if and onl if for each z L each point of the unit sphere with respect to z is an extremal point of the closed unit ball with respect to z. Proof. Let L be strictl convex space. It will be proven that each point of the set S z (0,1) is an extremal point of the set B z[0,1]. Let u S z (0,1) and let u tx (1 t) for some t (0,1) and some x, B z [0,1]. Thereb u S z (0,1), it is true that uz, 1 and since x, B z [0,1] we get that x, z 1,, z 1. It will be proven that x, z, z 1holds true. Indeed, in otherwise it holds that 1 u, z tx (1 t), z t x, z (1 t), z 1, (5)

7 A characterization of strictl convex -normed space 69 which is contradictor. Thus, x, z x, z, z. At the end, it will be proven that x, z holds. Indeed, in otherwise it holds that which is contradictor. So, 1 u, z tu (1 t) u, z t[ tx (1 t) ] (1 t)[ tx (1 t) ], z t x t(1 t)( x ) (1 t), z t t(1 t) (1 t) 1, x, z, z, z 1 and since L is strictl convex space, we get that, i.e. u is an extremal point of the closed unit ball with respect to z. Conversel, let s assume that x, z, z, z 1 and z, for x,, z L. Thus, the point u 1x 1 is on the unit sphere with respect to z. Therefore, it is an extremal point of the closed unit ball with respect to z, that is, i.e. L is strictl convex space. Example 3. Over the vector space C [0,1], of continuous functions on the interval [0,1], the function, : C[0,1] C[0,1] R defined as x( t) x( s) x, max st, [0,1] ( t) ( s). is a -norm. So, ( C [0,1],, ) is -normed space. The functions satisf x( t) 1, ( t) 1 t, z( t) t C [0,1] 1 1 x, z max max s t 1 and s, t [0,1] t s s, t [0,1] 1 t 1 s, z max max s t st ts 1. s, t [0,1] t s s, t [0,1] Therefore, x, S z (0,1). Further the functions u( t) 1x( t) 1 ( t) 1 t satisf 1 t 1 s 1 1 s, t [0,1] s, t [0,1] u, z max max s t ts st 1. t s

8 70 S. Malčeski, K. Anevska, R. Malčeski Therefore u S z (0,1) and since it is not an extremal point of B z[0,1] we deduce that the space ( C [0,1],, ) is not strictl convex space. In the purpose of the next characterization of strictl convex -normed space we will introduce the term of minimal point with respect to the set M L. The point v Lis said to be minimal point with respect to the set M if u m, z v m, z, z V ( u M) V({ u m m M}), u, z L and for each m M implies that u v. Theorem 5. -normed space L is strictl convex if and onl if for all x, L the points of the segment [ x, ] { tx (1 t) t [0,1]} are minimal with respect to the set { x, }. Proof. Let L be strictl convex space. Clearl, the points x and are minimal with respect to the set { x, }. Thus, let vt tx (1 t), t (0,1) be an point of the opened line segment [ x., ] Let s assume that for some u L and z u x, z vt x, z (1 t) x, z (5) u, z vt, z t x, z. (6) hold true. Further, the inequalities (5) and (6) impl x, z x u, z u x, z (1 t) x, z t x, z x, z. Thus, the following equalities hold true u x, z (1 t) x, z, u, z t x, z and since t (0,1) and z V ( x u, u), theorem 1 implies that u tx (1 t) vt. That is, v t is minimal point with respect to the set { x, }. Conversel, let for all a, b L the points of the segment [ ab, ] be minimal for the set { ab, }. If x, z, z, z 1 and z, then 0 x, z x, z, z 1x 1( ) x, z, 0 ( ), z, z, z 1x 1( ) ( ), z and z V ( x, ) V(0 x,0 ( )).

9 A characterization of strictl convex -normed space 71 But, the point 1 1 x ( ) belongs to a segment [ x, ]. That is, it is minimal with respect to the set { x, }. So, 1 1 ( ) 0, i.e.. That is, L is strictl convex. CONFLICT OF INTEREST No conflict of interest was declared b the authors. AUTHOR S CONTRIBUTIONS All authors contributed equall and significantl to writing this paper. All authors read and approved the final manuscript. References [1] Y. J. Cho, K. S. Ha and S. W. Kim, Strictl Convex Linear -Normed Spaces, Math. Japonica, Vol. 6 (1981), [] Y. J. Cho, B. H. Park and K. S. Park, Strictl -convex Linear -normed Spaces, Math. Japon. Vol. 7 (198), [3] C. Diminnie, S. Gähler and A. White, -Inner Product Spaces, Demonstratio Mathematica, Vol. VI, No 1 (1973), [4] C. Diminnie, S. Gähler and A. White, -Inner Product Spaces II, Demonstratio Mathematica, Vol. X, No 1 (1977), [5] C. Diminnie and S. Gahler, Some geometric results concerning strictl - convex -normed spaces, Math. Seminar Notes, Kobe Univ., Vol. 6 (1978), [6] C. Diminnie, S. Gahler and A. White, Strictl Convex Linear -Normed Spaces, Math. Nachr. Vol. 59 (1974), [7] C. Diminnie, S. Gahler and A. White, Remarks on Strictl Convex and Strictl -Convex -Normed Spaces, Math.Nachr. Vol. 88 (1979), [8] C. Diminnie and A. White, Strict convexit in topological vector spaces, Math. Japon. Vol. (1977), [9] R. Ehret, Linear -normed Spaces, Doctoral Diss., Saint Louis Univ., [10] R. W. Freese, Y. J. Choand S. S. Kim, Strctl -convex linear -normed spaces, J. Korean Math. Soc. Vol. 9 (199),

10 7 S. Malčeski, K. Anevska, R. Malčeski [11] S. Gähler, Lineare -normierte Räume, Math. Nachr. Vol. 8 (1965), 1-4. [1] K. S. Ha, Y. J. Cho and A. White, Strictl convex and strictl -convex linear - normed spaces, Math. Japon. Vol. 33 (1988), [13] K. Iseki, On non-expansive mappings in strictl convex -normed spaces, Math. Seminar Notes, Kobe Univ., Vol. 3 (1975), [14] K. Iseki, Mathematics in -normed spaces, Math. Seminar Notes, Kobe Univ., Vol. 6 (1976), [15] I. V. Istrăţescu, Strict convexit and complex strict convexit (theor and applications), Marcel Dekker, New York Basel, [16] A. Khan and A. Siddiqi, Characterization of Strictl Convex Linear -normed Spaces in Therms of Dualit Bimappings, Math. Japonica Vol. 3, No. 1, (1978). [17] S. Mabizela, A characterization of strictl convex linear -normed spaces, Questiones Math. Vol. 1 (1989), [18] A. Malčeski, l as а n-normed space, Matematički bilten, Vol. 1 (XXI) (1997), [19] A. Malčeski and R. Malčeski, L 1 (μ) as a n-normed space, Annuaire des l institut des mathematiques, Vol. 38 (1997), 3-9. [0] A. Malčeski and R. Malčeski, L p (μ) as a -normed space, Matematički bilten, Vol. 9 (XL) (005), [1] R. Malčeski, Strong convex n-normed spaces, Macedonian Academ of Sciences and Arts, Contributions, Vol. XVIII 1- (1997), [] R. Malčeski, Strong n-convex n-normed spaces, Matematički bilten, Vol. 1 (XXI) (1997), [3] R. Malčeski and K. Anevska, Strictl convex in quasi -pre-hilbert space, IJSIMR, e-issn , p-issn X, Vol., Issue 07 (014), [4] R. Malčeski, Lj. Nasteski, B. Načevska and A. Huseini, About the strictl convex and uniforml convex normed and -normed spaces, IJSIMR, e- ISSN , p-issn X, Vol., Issue 06 (014), [5] A. White, -Banach Spaces, Math. Nachr. Vol. 4 (1969), ) Universit Euro-Balkan, Skopje, Macedonia address: samoil.malcheski@gmail.com ) Facult of informatics, FON Universit, Skopje, Macedonia address: risto.malceski@gmail.com 3) Facult of informatics, FON Universit, Skopje, Macedonia address: anevskak@gmail.com