International Journal of Mathematical Analysis 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 space 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-inner product space which is not 2-pre-Hilbert space. Further, more properties of 2*-semi-inner product are given, and also is proved that 2*-inner product defines 2-semi-norm.
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 vector space with dimension 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 {}ii1, aLi , i 1,2,3,... such that 1/ p( a , b ) ( || aii , b || ) , for every a,() b l L . i1 Clearly, lL () with the operations addition and multiplying by real number, 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 inequality 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 ||)] ii11 1/ 1/ ( ||,ai c i ||) ( ||, b i c i ||) ii11 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 ii1 i i1 a p(,')( a a ||, a i ||)1/ 0 , i i i1 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 LLLsuch 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 0a tba , tbc | aac , | 2 tabc , | t2 bbc , | , which implies the inequality 4a , 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 . i1 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 . i1 i , j 1 ij 2608 Samoil Malčeski et al.
Again applying statements a), c), d) and Corollary 1 we have
kk1 a1, a 2 |i a i a 1 , a 2 | i a i k 1 a k 1 ii11 k a1, a 2 | i a i a 1 , a 2 | k 1 a k 1 i1 kk 1 [ a , a | a a a , a | a a ] 2 i i k1 k 1 1 2 i i k 1 k 1 1 2 ii11 k 2 i j a1, a 2 | a i a j k 1 a 1 , a 2 | a k 1 ij,1 ij k ki1 aa 1, 2 | aaik1 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 k1 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