2-Semi-Norms and 2*-Semi-Inner Product

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1 International Journal of Mathematical Analysis Vol. 8, 01, no. 5, HIKARI Ltd, -Semi-Norms and *-Semi-Inner Product Samoil Malčesi Centre for research and development of education Sope, Macedonia Risto Malčesi Faculty of informatics, FON University Bul. Vovodina bb 1000 Sope, Macedonia Katerina Anevsa Faculty of informatics, FON University Bul. Vovodina bb 1000 Sope, Macedonia Copyright 01 Samoil Malčesi, Risto Malčesi and Katerina Anevsa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. Abstract The Cartesian product of pre-hilbert spaces ( L1,(, ) 1) and ( L,(, ) ) is pre-hilbert space in which the inner product is defined by (( a1, b1 ),( a, b )) ( a1, a) 1 ( b1, b ), ( a1, b1 ),( a, b ) L1 L. In this paper is proved that analogously construction doesn t hold for -pre- Hilbert spaces, and thus is given the term *-semi-inner product. Also, is proved the existence of *-semi-inner product space which is not -pre-hilbert space. Further, more properties of *-semi-inner product are given, and also is proved that *-inner product defines -semi-norm. 010 Mathematics Subect Classification. 6B0, 6C05 Keywords: inner product, -semi-norm, -inner product, *-semi-inner product

2 60 Samoil Malčesi et al. 1. Introduction Let L be a real vector space with dimension greater than 1 and, be a real function on L L satisfying: a) ab, 0 b L and ab, 0 if and only if the set { ab, } is linearly dependent; b) a, b b, a b L ; c) a, b a, b b L and for every R, d) a b, c a, c b, c b, c L. Function, is called as -norm on L, and ( L,, ) is called as linear -normed space ([]). Let n 1 be a natural number, L be a real vector space, diml n and (, ) be a real function on LL L such that: i) ( a, a b) 0 b L и ( a, a b) 0 if and only if a and b are linearly dependent; ii) ( a, b c) ( b, a c) b, c L ; iii) ( a, a b) ( b, b a) b L ; iv) ( a, b c) ( a, b c) b, c L and for every R ; and v) ( a a1, b c) ( a, b c) ( a1, b c) b, a1, c L. Function (, ) is called as -inner product, and ( L,(, )) is called as -pre- Hilbert space ([1]). Concepts of -norm and -inner product are two dimensional analogies of the concepts of norm and inner product. R. Ehret proved ([3]) that, if ( L,(, )) be -pre-hilbert space, than 1/ a a b (1) a, b (, ) defines -norm. So, we get vector -normed space ( L,, ). In -normed and -pre-hilbert spaces go properties analogous to properties of normed and pre- Hilbert spaces. But, the last state is not always correct, and in this paper we ll show that.. Notes for -Semi-Norms If ( L1,(, ) 1) and ( L,(, ) ) be pre-hilbert spaces, then (( a1, b1 ),( a, b )) ( a1, a) 1 ( b1, b ), for every ( a1, b1 ),( a, b ) L1 L

3 -Semi-norms and *-semi-inner product 603 defines an inner product, and thus, L1 L is pre-hilbert space, i.e. normed space in which the norm is defined by ( a, b) ( a, a) 1 ( b, b), for every ( a, b) L1 L. But, if ( L1,(, ) 1) and ( L,(, ) ) are -pre-hilbert spaces, then ( a1, b1 ),( a, b ) ( a3, b3 ) ( a1, a a3) 1 ( b1, b b3), () for ( a1, b1 ),( a, b ),( a3, b3) L1 L is not defined -inner product on L1 L. Really, it s easy to chec that () defines a real function on ( L1 L ) ( L1 L ) ( L1 L ) which satisfies the Axioms ii), iii), iv) and v) of Definition of -inner product. Further, if ( a1, b 1) and ( a, b ) are linearly dependent, then exists R such that ( a, b ) ( a, b ) ( a, b ). 1 1 It actually means a1 a, b1 b, i.e. a 1 and a are linearly dependent in L 1, and b 1 and b are linearly dependent in L. Hence, ( a, b ),( a, b ) ( a, b ) ( a, a a ) ( b, b b ) But, the vectors ( ab, ) and ( a, b ) are not linearly dependent in L1 L. So, for these vectors we have ( a, b),( a, b) ( a, b) ( a, a a) ( b, b b) 0, 1 So, the Axiom i) of Definition of -inner product doesn t hold. The already said imply that function () does not define -norm of L1 L, as by (1) can be done in -pre-hilbert spaces ( L1,(, ) 1) and ( L,(, ) ). But the function p : ( L1L ) R defined by p(( a, b ),( a, b )) ( a, b ),( a, b ) ( a, b ) , (3) for every ( a1, b1 ),( a, b ) L1 L holds the conditions 1) ) of the following Definition. Definition 1 ([1]). Let L be a real vector space with dimension greater than 1 and p : LL R be a function such that: 1) If a, b L are linearly dependent, then p( a, b) 0, ) p( a, b) p( b, a) b L,

4 60 Samoil Malčesi et al. 3) p( a, b) p( a, b) b L and for every R, ) p( a b, c) p( a, c) p( b, c) b, c L. The function p is called as -semi-norm, and ( Lp, ) is called as -semi-normed space. Clearly, each -norm is -semi-norm, but not each -semi-norm is -norm. Further, each -semi-norm satisfies the following properties. Lemma 1 ([1]). Let ( Lp, ) be a -semi-normed space. Then a) p( a, b) p( a b, b) b L and for every R, b) p( a b, a b) p( a, b) b L and for every,, R, c) for every a L the function p 1 ( x) p( x, a), for every x L is semi- norm on L. Theorem 1 ([1]). Let ( Lp, ) be a -semi-normed space. Then, a) p( a, c) p( b, c) p( a b, c) b, c L, b) p( a, b) 0 b L, c) for every a L the set { x p( x, a) 0} is subspace of L. Example 1. Let ( L,, ) be a -normed space, 1 and l ( L) denotes the set of all sequences a { a i } i 1, ai L, i 1,,3,... such that 1/ p( a, b) ( ai, bi ) b l ( L). i1 Clearly, l ( L) with the operations addition and multiplying by real number, defined as in above of space l ( L) is real vector space. It s easy to note that function p holds the conditions 1), ) and 3) of Definition 1. Further, by parallelepiped inequality and Minowsi inequality follows b, c L is true the following: 1/ 1/ i i i i i i i i1 i1 1/ 1/ ai ci bi ci i1 i1 p( a b, c) ( a b, c ) [ ( a, c b, c ) ] (, ) (, ) p( a, c) p( b, c), So, p is -semi-norm on l ( L). But, p is not -norm on l ( L), because if

5 -Semi-norms and *-semi-inner product 605 a a { ai} i 1 l ( L), then is easy to notice that ' { i a } i 1 l ( L) i and holds a 1/ (, ') (, i p a a ai ) 0, i i1 but the set { aa, '} is not linearly dependent in l ( L). 3. *-Semi-Inner Product The mentioned above, is a direct reason of giving the following definition i.e. of giving the term *-semi-inner product, which is used for generating -semi-norm, analogously as -inner product generates -norm. Definition. Let L be a real vector space with dimension greater than 1 and, be a real function on LLLsuch that 1. If a, b L are linearly dependent, than a, a b 0,. a, b c b, a c b, c L, 3. a, a b b, b a b, c L,. a, b c a, b c b, c L and for every R, 5. a a1, b c a, b c a1, b c b, a1, c L. Function, is called as *-semi-inner product, and ( L,, ) is called as space with *-semi-inner product. Example. Let ( L1,(, ) 1) and ( L,(, ) ) are pre-hilbert spaces and i, i 1, is the norm of inner product (, ) i. By Corollary 1, [], ( a, b) i ( a, c) i ( a, b c) i ( a, b) i c i ( a, c) i ( b, c) i ( c, b) i ( c, c), i defines -inner product on L i. Thus, we get -pre-hilbert spaces ( L1,(, ) 1) and ( L,(, ) ). Further, on L1 L, () defines *-semi-inner product, and (3) defines -semi-norm on L1 L. Letting ( L1,(, ) 1) ( L,(, ) ) ( L,(, )), we get that for each pre-hilbert space ( L,(, )) equality () defines *-semi-inner product on vector space L L, which is not -inner product. In this case () defines -semi-norm, which isn t norm. Theorem. Let ( L,, ) be a space with *-semi-inner product. For every a, b, c L the following inequality is satisfied

6 606 Samoil Malčesi et al. a, b c a, a c b, b c. () The inequality () is -dimensional analogy of Cauchy-Bunyaovsy-Schwarz inequality into space with *-inner product. Proof. By stated above, follows that the function p : LL R defined by p( a, b) a, a b b L is -semi norm. Now, by Theorem 1 b), is true that a, a b p( a, b) 0 b L, and thus, for every t R holds 0 a tb, a tb c a, a c ta, b c t b, b c, which implies the inequality which is equivalent to inequality (). a, b c b, b c a, a c 0, Corollary 1. For every a, c L holds a, c c a, c a 0. Proof. Directly is implied by Theorem and Axiom i) of definition. Lemma. Let ( L,, ) be a space with *-semi-inner product. а) for every a, b, c L and for every R holds a, b c a, b c. b) for every a, b, c, c' L holds a, b c c' a, b c c' c, c' a b c, c' a b. c) for every a, b, c, c' L holds a, b c c' a, b c a, b c' 1 [ c, c' a b c, c' a b ]. d) if a, b c a, b c' 0, then a, b c c' a, b c c'. e) for every a1, a,..., an L, n such that ai, a a 0, за i i and for every real numbers 1,,..., n holds n n a1, a iai i a1, a ai a i1 i, 1 i Proof. а) We have a, b c 1 [ a b, a b c a b, a b c] 1 [ c, c a b c, c a b] b) We have. [ c, c a b c, c a b] [ a b, a b c a b, a b c] a, b c.

7 -Semi-norms and *-semi-inner product 607 a, b c c' a, b c c' 1 [ a b, a b c c ' a b, a b c c ' a b, a b c c ' a b, a b c c ' ] 1 [ c c', c c' a b c c ', c c ' a b c c', c c' a b c c ', c c ' a b] c, c' a b c, c' a b. c) We have a, b c c' 1 [ a b, a b c c ' a b, a b c c ' ] 1 [ c c', c c' a b c c ', c c ' a b] 1 [ c, c a b c', c' a b c, c' a b c, c a b c', c' a b c, c' a b] 1 [ a b, a b c a b, a b c a b, a b c ' a b, a b c' ] 1 [ c, c' a b c, c' a b] a, b c a, b c' 1 [ c, c' a b c, c' a b]. d) By statements a) and c) we have a, b c c ' a, b c a, b c ' 1 [ c, c ' a b c, c ' a b] a, b c a, b c ' 1 [ c, c' a b c, c' a b] 1 [ c, c' a b c, c' a b] { a, b c a, b c ' 1 [ c, c' a b c, c' a b]} a, b c c '. e) The statement will be proven by induction. By statements a), c), d) and Corollary 1 we have a1, a 1a1 a 1 a1, a a1 a1, a a 1 [ a 1, a a1 a a1, a a1 a] 1 a1, a a1 a. Assume 1, i i i i1 i, 1 1, i i, for a a a a a a a.

8 608 Samoil Malčesi et al. Again applying statements a), c), d) and Corollary 1 we have 1 a1, a iai a1, a iai 1a 1 i1 i1 a1, a iai a1, a 1a 1 i1 1 [, iai 1a 1 a1 a iai, 1a 1 a1 a] i1 i1 i a1, a ai a 1a1, a a 1 i, 1 i 1 ia1, a ai a1 i, 1 i a1, a ai a 1 ia1, a ai a 1 i, 1 i, 1 i 1 i a1, a ai a. i, 1 i References [1] C. Diminnie, S. Gähler and A. White, -Inner Product Spaces, Demonstratio Mathematica, Vol. VI, (1973), [] C. Diminnie, S. Gähler and A. White, -Inner Product Spaces II, Demonstratio Mathematica, Vol. X, No 1, (1977), [3] R. Ehret, Linear -Normed Spaces, Doc. Diss., Saint Louis Univ., [] S. Gähler, Lineare -normierte Räume, Math. Nachr., 8, (1965), [5] S. Gähler and A. Misia, Remars on -Inner Product, Demonstratio Mathematica, Vol. XVII, No 3, (198),

9 -Semi-norms and *-semi-inner product 609 [6] A. Malčesi, l as n-normed space, Мatematiči bilten, V. 1, (1997), [7] A. Malčesi and R. Malčesi, L 1 (μ) as a n-normed space, Annuaire des l institut des mathematiques, V. 38, (1997), 3-9. [8] A. Malčesi and R. Malčesi, L p (μ) as a -normed space, Мatematiči bilten, V 9, (005), [9] R. Malčesi, Remars on n-normed spaces, Matematiči bilten, V. 0, (1996), 33-50, (in macedonian) [10] R. Malčesi, Strong convex n-normed spaces, Macedonian Academy of Sciences and Arts, Contributions, XVIII 1-, (1997), [11] R. Malčesi, Strong n-convex n-normed spaces, Мatematiči bilten, V 1, (1997), pp [1] R. Malčesi and A. Malčesi, n-seminormed space, Annuaire des l institut des mathematiques, V. 38, (1997), 31-0, (in macedonian) Received: October 9, 01; Published: November 0, 01