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International Journal of Vol. 8, 2014, no. 52, 2601 - 2609 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.410322

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

Samoil Malčeski

Centre for research and development of education Skopje, Macedonia

Risto Malčeski

Faculty of informatics, FON University Bul. Vojvodina bb 1000 Skopje, Macedonia

Katerina Anevska

Faculty of informatics, FON University Bul. Vojvodina bb 1000 Skopje, Macedonia

Copyright © 2014 Samoil Malčeski, Risto Malčeski and Katerina Anevska. 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 work is properly cited.

Abstract

The Cartesian product of pre-Hilbert spaces (,(,))L11 and (,(,))L22 is pre-Hilbert in which the inner product is defined by ((,),(a1 b 1 a 2 , b 2 )) (, a 1 a 2 ) 1 (, b 1 b 2 ) 2 , (a1 , b 1 ),( a 2 , b 2 ) L 1 L 2 . In this paper is proved that analogously construction doesn’t hold for 2-pre- Hilbert spaces, and thus is given the term 2*-semi-inner product. Also, is proved the existence of 2*-semi- which is not 2-pre-. Further, more properties of 2*-semi-inner product are given, and also is proved that 2*-inner product defines 2-semi-.

2010 Mathematics Subject Classification. 46B20, 46C05

Keywords: inner product, 2-semi-norm, 2-inner product, 2*-semi-inner product

2602 Samoil Malčeski et al.

1. Introduction

Let L be a real with greater than 1 and || , || be a real function on LL satisfying: a) ||ab , || 0 , for every a, b L and ||ab , || 0 if and only if the set {,}ab is linearly dependent; b) ||a , b || || b , a ||, for every a, b L ; c) ||a , b || | | || a , b ||, for every a, b L and for every  R, d) ||a b , c ||  || a , c ||  || b , c ||, for every a,,. b c L Function || , || is called as 2-norm on L, and (L ,|| , ||) is called as linear 2-normed space ([4]).

Let n 1 be a natural number, L be a real vector space, dimLn and ( ,  |  ) be a real function on LLL such that: i) (a , a | b ) 0 , for every a, b L и (a , a | b ) 0 if and only if a and b are linearly dependent; ii) (a , b | c ) ( b , a | c ) , for every a,, b c L ; iii) (a , a | b ) ( b , b | a ) , for every a, b L ; iv) (a , b | c ) ( a , b | c ) , for every a,, b c L and for every  R ; and v) (a a11 ,|)(,|)(,|) b c  a b c  a b c , for every a,,, b a1 c L .

Function ( ,  |  ) is called as 2-inner product, and (L ,( ,  |  )) is called as 2-pre- Hilbert space ([1]). Concepts of 2-norm and 2-inner product are two dimensional analogies of the concepts of norm and inner product. R. Ehret proved ([3]) that, if (L ,( ,  |  )) be 2-pre-Hilbert space, than ||a , b || ( a , a | b )1/2 (1)

defines 2-norm. So, we get vector 2-normed space (L ,|| , ||) . In 2-normed and 2-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.

2. Notes for 2-Semi-Norms

If (,(,))L11 and (,(,))L22 be pre-Hilbert spaces, then ((,),(a1 b 1 a 2 , b 2 )) (, a 1 a 2 ) 1 (, b 1 b 2 ) 2 , for every (a1 , b 1 ),( a 2 , b 2 ) L 1 L 2

2-Semi-norms and 2*-semi-inner product 2603

defines an inner product, and thus, LL12 is pre-Hilbert space, i.e. normed space in which the norm is defined by

||(,)||a b (,) a a12 (,) b b , for every (,)a b L12 L . But, if (L11 ,( ,  |  ) ) and (L22 ,( ,  |  ) ) are 2-pre-Hilbert spaces, then

(,),(,)|(,)abab11 22 ab 33   (, aaa 1231 | )  (, bbb 1232 |) , (2)

for (a1 , b 1 ),( a 2 , b 2 ),( a 3 , b 3 ) L 1 L 2 is not defined 2-inner product on LL12 . Really, it’s easy to check that (2) defines a real function on ()()()LLLLLL1 2  1  2  1  2 which satisfies the Axioms ii), iii), iv) and v) of Definition of 2-inner product. Further, if (,)ab11 and (,)ab22 are linearly dependent, then exists  R such that

(,)(,)(,)a1 b 1 a 2 b 2  a 2  b 2 . It actually means a1 a 2, b 1 b 2 , i.e. a1 and a2 are linearly dependent in L1 , and b1 and b2 are linearly dependent in L2 . Hence, (,),(,)|(,)abab11 11 ab 22   (, aaa 1121 | )  (,|) bbb 1122  0 . But, the vectors (,)ab and (ab ,2 ) are not linearly dependent in LL12 . So, for these vectors we have

(,),(,)|(,2)abab ab   (, aaa |)12  (,|2) bbb  0 , So, the Axiom i) of Definition of 2-inner product doesn’t hold. The already said imply that function (2) does not define 2-norm of LL12 , as by (1) can be done in 2-pre-Hilbert spaces (,(,))L11 and (,(,))L22 . But the 2 function p:() L12 L R defined by

pabab((1 , 1 ),( 2 , 2 ))  ( ababab 1 , 1 ),( 1 , 1 )|( 2 , 2 )  , (3)

for every (a1 , b 1 ),( a 2 , b 2 ) L 1 L 2 holds the conditions 1) – 4) of the following Definition.

Definition 1 ([12]). Let L be a real vector space with dimension greater than 1 and p: L L R be a function such that: 1) If a, b L are linearly dependent, then p( a , b ) 0 , 2) p(,)(,) a b p b a , for every a, b L , 2604 Samoil Malčeski et al.

3) p( a , b ) | | p ( a , b ), for every a, b L and for every  R , 4) p(,)(,)(,) a b c  p a c  p b c , for every a,, b c L . The function p is called as 2-semi-norm, and (,)Lp is called as 2-semi-normed space. Clearly, each 2-norm is 2-semi-norm, but not each 2-semi-norm is 2-norm. Further, each 2-semi-norm satisfies the following properties.

Lemma 1 ([12]). Let (,)Lp be a 2-semi-normed space. Then a) p(,)(,) a b p a b b , for every a, b L and for every  R , b) p( a  b ,  a   b )  |    | p ( a , b ) , for every a, b L and for every ,, , R c) for every aL the function p1()(,) x p x a , for every xL is semi- norm on L . ■

Theorem 1 ([12]). Let (,)Lp be a 2-semi-normed space. Then, a) |(,)p a c p (,)| b c  p ( a  b ,) c , for every a,, b c L , b) p( a , b ) 0 , for every a, b L , c) for every aL the set {x | p ( x , a ) 0} is subspace of L . ■

Example 1. Let (L ,|| , ||) be a 2-normed space, 1   and lL () denotes  the set of all sequences aa {}ii1, aLi  , i 1,2,3,... such that  1/  p( a , b ) ( || aii , b || )   , for every a,() b l L . i1 Clearly, lL () with the operations addition and multiplying by , defined as in above of space lL () is real vector space. It’s easy to note that function p holds the conditions 1), 2) and 3) of Definition 1. Further, by parallelepiped and Minkowski inequality follows, for every a,, b c L is true the following:

 1/   1/  pabc( ,)(  || abci  i , i ||)  [ (||, ac i i ||||,  bc i i ||)] ii11  1/   1/  ( ||,ai c i ||) ( ||, b i c i ||) ii11 p( a , c ) p ( b , c ),

So, p is 2-semi-norm on lL (). But, p is not 2-norm on lL (), because if 2-Semi-norms and 2*-semi-inner product 2605

a a{}() a l L , then is easy to notice that a'{}()i  l L and holds ii1 i i1  a p(,')( a a ||, a i ||)1/ 0 ,  i i i1 but the set {aa , '} is not linearly dependent in lL (). ■

3. 2*-Semi-Inner Product

The mentioned above, is a direct reason of giving the following definition i.e. of giving the term 2*-semi-inner product, which is used for generating 2-semi-norm, analogously as 2-inner product generates 2-norm.

Definition 2. Let L be a real vector space with dimension greater than 1 and ,|   be a real function on LLLsuch that

1. If a, b L are linearly dependent, than a, a | b   0 , 2. a, b | c    b , a | c  , for every a,, b c L , 3. a, a | b    b , b | a  , for every a,, b c L , 4. a, b | c    a , b | c  , for every a,, b c L and for every  R , 5. a  a11, b | c    a , b | c    a , b | c  , for every a,,, b a1 c L . Function ,|   is called as 2*-semi-inner product, and (L , ,  |  ) is called as space with 2*-semi-inner product.

Example 2. Let (,(,))L11 and (,(,))L22 are pre-Hilbert spaces and || ||i , i  1,2 is the norm of inner product (,)i . By Corollary 1, [2], (,)(,)a bii a c 2 (,|)a b ci  (,)||||(,)(,) a b i c i  a c i  b c i , (,)(,)c bii c c defines 2-inner product on Li . Thus, we get 2-pre-Hilbert spaces (L11 ,( ,  |  ) ) and (L22 ,( ,  |  ) ) . Further, on LL12 , (2) defines 2*-semi-inner product, and (3) defines 2-semi-norm on LL12 . Letting (LLL1 ,(,))  1  ( 2 ,(,))   2  (,(,))   , we get that for each pre-Hilbert space (L ,( , )) equality (2) defines 2*-semi-inner product on vector space LL , which is not 2-inner product. In this case (2) defines 2-semi-norm, which isn’t norm. ■

Theorem 2. Let (L , ,  |  ) be a space with 2*-semi-inner product. For every a,, b c L the following inequality is satisfied 2606 Samoil Malčeski et al.

|a , b | c  |   a , a | c   b , b | c  . (4)

The inequality (4) is 2-dimensional analogy of Cauchy-Bunyakovsky-Schwarz inequality into space with 2*-inner product. Proof. By stated above, follows that the function p: L L R defined by p( a , b )  a , a | b  , for every a, b L is 2-semi norm. Now, by Theorem 1 b), is true that a, a | b   p ( a , b )  0 , for every a, b L , and thus, for every t R holds 0a tba ,  tbc |  aac , |  2 tabc , |  t2 bbc , |  , which implies the inequality 4a , b | c 2  4  b , b | c   a , a | c   0 , which is equivalent to inequality (4). ■

Corollary 1. For every a, c L holds a, c | c    a , c | a   0 . Proof. Directly is implied by Theorem 2 and Axiom i) of definition 2. ■

Lemma 2. Let (L , ,  |  ) be a space with 2*-semi-inner product. а) for every a,, b c L and for every  R holds a, b | c  2  a , b | c  . b) for every a,,,' b c c L holds abcc,|  ' abcc ,|  ' ccab ,'|  ccab ,'|  . c) for every a,,,' b c c L holds abcc,|  ' abc ,|  abc ,|' 1 [,'| ccab  ccab ,'|  ] . 2 d) if a, b | c    a , b | c '   0, then a, b | c  c '    a , b | c  c '  . e) for every a12, a ,..., an  L , n  2 such that ai, a j | a k   0 , за i j k

 i and for every real numbers 12,  ,..., n holds nn a1, a 2 |i a i    i  j  a 1 , a 2 | a i  a j  . i1 i , j 1 ij Proof. а) We have abc, | 1 [ ababc ,  |   ababc ,  |   ] 4 1 [ c ,  c | a  b     c ,  c | a  b  ] 4 2  [ c , c | a  b    c , c | a  b  ] 4 2  [ ababc ,  |  ababc ,  |  ]2 abc , |  . 4 b) We have

2-Semi-norms and 2*-semi-inner product 2607

abcc,|',|'[,|'  abcc 1 ababcc   ababcc ,|'   4 ababcc,  |  ' ababcc ,  |  ' ] 1 [ccccab ',  ' |  ccccab ',  ' |  4 ccccab',  ' |  ccccab ',  ' |  ]  c, c ' | a  b    c , c ' | a  b  . c) We have

abcc,|'[,|' 1 ababcc   ababcc ,|']   4 1 [ccccab ',  ' |  ccccab ',  ' |  ] 4 1 [,|ccab  ccab ','|  2,'| ccab  4 ccab,|  ccab ','|  2,'| ccab  ] 1 [ ababc ,  |  ababc ,  |  ababc ,  | '  aba , bc| '  ]  4 1 [ c , c ' | a  b    c , c ' | a  b  ] 2 abc,|  abc ,|' 1 [,'| ccab  ccab ,'|  ]. 2 d) By statements a) and c) we have abcc,|  ' abc ,|  abc ,|' 1 [,'| ccab  ccab ,'|  ] 2 abc,|  abc ,|' 1 [,'| ccab  ccab ,'|  ] 2  1 [ c , c ' | a  b    c , c ' | a  b  ] 2 {,|abc  abc ,|' 1 [,'| ccab  ccab ,'|  ]} 2  a, b | c  c '  . e) The statement will be proven by induction. By statements a), c), d) and Corollary 1 we have 22 a1211, a | a   22 a    1121  a , a | a    2122  a , a | a    12[ a , a | a  a    a , a | a  a  ] 2 1 2 1 2 1 2 1 2 1 2 a 1, a 2 | a 1  a 2  . Assume kk a1, a 2 |i a i    i  j  a 1 , a 2 | a i  a j  , for k  2 . i1 i , j 1 ij 2608 Samoil Malčeski et al.

Again applying statements a), c), d) and Corollary 1 we have

kk1 a1, a 2 |i a i    a 1 , a 2 |  i a i   k 1 a k 1  ii11 k  a1, a 2 | i a i    a 1 , a 2 | k 1 a k 1   i1 kk 1 [ a ,  a | a  a     a ,  a | a  a  ] 2 i i k1 k  1 1 2 i i k  1 k  1 1 2 ii11 k 2  i  j a1, a 2 | a i  a j    k 1  a 1 , a 2 | a k 1   ij,1 ij k ki1 aa 1, 2 | aaik1 i,1 kk i  j a1, a 2 | a i  a j    k 1  i  a 1 , a 2 | a i  a k 1  i, j 1 i , 1 ij k1  i j a12, a | a i  a j  . ■ ij,1 ij

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2-Semi-norms and 2*-semi-inner product 2609

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Received: October 29, 2014; Published: November 20, 2014