ASYMMETRIC INNER PRODUCT AND THE ASYMMETRIC QUASI NORM FUNCTION. Stela Çeno 1, Gladiola Tigno 2

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1 Математички Билтен ISSN X (print) Vol 4(LXVI) No 3 ISSN (online) 16 (61-67) UDC: Скопје, Македонија ASYMMETRIC INNER PRODUCT AND THE ASYMMETRIC QUASI NORM FUNCTION Stela Çeno 1, Gladiola Tigno Abstract This paper attempts to generalize the semi scalar product concept according to G Lumer by replacing Cauchy inequality with another inequality which is more generalized Based on this attempt of generalization it is built a function which fulfils the conditions which are changed In this paper it is also generalized quasi norm function by replacing homogeneity condition with a more restricted condition by producing this time a more generalized asymmetric semi norm function As a result, in this paper it is defined the asymmetric inner product function and the asymmetric quasi norm function Moreover, it is even given relation between these two 1 INTRODUCTION Semi-inner products, that can be naturally defined in general Banach spaces over the real or complex number field, play an important role in describing the geometric properties of these spaces Starting from its axiomatic, many researchers have made various modifications passing in its generalization Semi-scalar products mark the very first generalizations of the scalar product function The strong bond between these functions with the norm function has made it possible to obtain a lot of interesting results which are connected with the orthogonality and convexity [1],[] In [3],[4] it is also generalized the quasi norm function by replacing homogeneity condition with a more restricted condition by producing this time a more generalized asymmetric semi norm function Let be p : a function defined by: x, x p ( x) x, x 1 Mathematics Subject Classification 46B Key words and phrases Quasi norm function, asymmetric quasi inner product, asymmetric quasi norm function

2 6 Stela Çeno 1, Gladiola Tigno Definition 1 The p : a) p ( b) p( x) p( x) for, x c) p ( x y) p( x) p( y) xy, function is called an asymmetric semi norm if: For every x( x1, x), we define the function p( x) p ( x1 ) p( x), where p ( x1 ), p ( x ) are asymmetric semi norms in Proposition 1 The function p : such that p( x) p ( x1 ) p( x) it is also an asymmetric semi norm in Proof a) We have b) We have c) We have p( x) p ( x1 ) p( x), ( x1, x) and p( x) p( x1 ) p( x) x1 x p( x) p ( x1 ) p ( x) p ( x1 ) p ( x) [ p ( x1 ) p( x)] p( x), for p( x y) p ( x y ) p ( x y ) So p( x y) p( x) p( y), 1 1 p ( x ) p ( y ) p ( x ) p ( y ) 1 1 [ p ( x ) p ( x )] [ p ( y ) p ( y )] 1 1 p( x) p( y) xy, For every two points x( x1, x ) and y( y1, y ) in (, ): such that: x1 y1 xy p( y)[ ], for y1 and y, p ( y1 ) p ( y) xy 1 1 p( y), for y1 and y, ( xy, ) p( y1) xy p( y), for y1 and y, p( y), for y1 and y The function defined above have the following properties: we build the function 1) ( x, x), ( x1, x) ) For (, ) ( )[ x 1( y 1) x ( ) 1 1) x y p y y ( ) ( ) ] p ( y )[ p y p y p ( y ) p ( y ) ] 1 1

3 Asymmetric Inner Product and 63 py ) 1 1 ( )[ ] p ( y1 ) p ( y) ( x ) y ( x ) y ( x, y ) p ( y )[ ( 1) ( ) ] p ( y )[ p y p y p ( y1 ) p ( y) ] ( x, y ), 3) ( x x, y) ( x, y) ( x, y) Case 1 x1, Case x1 : In this case x ( x1, x), x ( x1, x) x : and y ( y1, y) where x1, x, ( x x ) y ( x x ) y ( x x, y) p( y)[ ] p ( y ) p ( y ) py ( )[ ] p ( y ) p ( y ) p ( y ) p ( y ) p( y)[ ] p( y)[ ] p ( y ) p ( y ) p ( y ) p ( y ) ( x, y) ( x, y) x ( x1, x), x ( x1,) 1 1 and y ( y1, y) where x1, x, 1 1 ( x, y) p( y)[ ] and p ( y1 ) p ( y) x x ( x1 x1, x) therefore: ( x1 x1 ) y1 x x p y p ( y1 ) p ( y) (, ) ( )[ ] 1 1 p( y1) ( x, y ) p ( y ) xy p y p ( y ) p ( y ) p ( y ) p( y)[ ] ( ) 1 1 ( x, y) ( x, y) The reconciliation ( x x, y) ( x, y) ( x, y) goes equally in these cases: a) b) x ( x1, x), x (, x) and y ( y1, y) where x1, x, x x ( x1,), x ( x1, x) and y ( y1, y) where x1, x1, x x (, x), x ( x1, x) and y ( y1, y) where x, x1, x x ( x, x ), x ( x, x ) y ( y, y ) x, x, c) Case 3: 1 1 and 1 where 1 x1, x but x1x1 and x x so x1 x1 and In this case xx (,) therefore ( x x, y) while: ( x, y) p( y)[ p( y)[ ] p ( y ) p ( y ) p ( y ) p ( y ) 1 1 p y 1 1 ( )[ ] ( x, y) p ( y1 ) p ( y) from where: ( x, y) ( x, y) ( x x, y) x x

4 64 Stela Çeno 1, Gladiola Tigno Case 4: x1, In this case while: and x ( x1, x), x ( x1, x) and y ( y1, y) where x1, x, x1x1 so x1 x1 x x (, x x) therefore: (, ) ( ) ( x ) x p y p ( ) ( y ) p ( ) ( y ) p y p y p ( y) x but 1 1 ( x, y) p( y)[ ] p ( y1 ) p ( y) p y x p y p ( y1 ) p ( y) p( y1) p( y) x x p ( y) p ( y) x ( x1, x), x ( x1, x) 1,, 1, but x x so x x (, ) ( )[ ] ( )[ ] Since, from ( x, y) ( x, y) p( y) p( y) ( x x, y) It is equally demonstrated when: where x x x x 4) From the definition of the function and y ( y1, y) x, x p ( x) x, x x p ( x), x we obtain the inequality:, from where: x1 p ( x1 ) x p( x), y1 p( y1) y p( y) brings: y1 y ( x, y) p( y)[ x1 x ] p ( y1 ) p( y) p( y)[ x1 x ] p( y) p( x1 ) p( x) p( y) p( x) p( x) p( y) So ( x, y) p( x) p( y), from where ( x, x) ( x, x) p ( x) Remark For ( xx, ) where x( x1, x) we have: 1) x1, x p( x1 ), p( x) x1 x x1 x p ( x1 ) p ( x) p ( x1 ) p ( x) p( x) p ( x1 ) p( x) p ( x) ) x1, x p( x1 ), p( x) ( ( ) x1 p x p 1 ( x1) p ( p ( 3) x1, x p( x1 ), p( x) ( x, x) p( x)[ ] p( x)[ ]

5 Asymmetric Inner Product and 65 ( x, x) ( ) x p x p ( x) x p ( x) p ( x) 4) x1, x p( x1 ) p( x) p( x) Finally: ( x, x) p ( x) x, x p ( x) Remark Frankly, ( x, x) p ( x) every time is not true Because for x ( 1,) we have p( x) 1 5 p ( x) 5 and other side: x1 x ( 1) p ( x1 ) p ( x) 1 ( x, x) p( x)[ ] 5[ ] 5[1 1] 1 In this case ( x, x) p ( x) Record 1 Also we can prove that Proof Case 1: For p ( x) ( x, x) x( x1, x) where x1 x we have: ( 1 1 x, x ) p ( x )[ x x 1 ( 1) ( ) ] p ( x )[ x x p x p x x1 x ] p ( x )[ x x ] p ( x ) p ( x ) p ( x) ( x, x) ( x, x) or Case : For ( 1, ) x x we have: x 1 x 1 1 p ( ) p ( x1 ) p ( x) x1 x x x x where 1 ( x, x ) p ( x )[ ] p ( x )[ ] p ( x )[ ] So p ( x) ( x, x) Case 3: For So p ( x) ( x, x) x( x1, x) and x1 x [ x1 x ] we have: 1 1 x x x x ( x, x) p( x)[ ] p( x)[ ] p ( x ) p ( x ) x x 1 1 x1 x1 x p ( x) ( )[ ] ( )[ ] p x x p x Record The function ( xy, ) defined as above provides the benefit of the function p : such that: p( x) ( x, x) From the inequality: p ( x) ( x, x) we have p ( x) p ( x) or p ( x) p ( x), and from the inequality ( x, y) p( x) p( y) we have: ( x, y) p( x) p( y) p ( x) p( y) p( x) p( y) So for the function p : these properties hold:

6 66 Stela Çeno 1, Gladiola Tigno 1) p( x), p( x) x for ) p( x) p( x), for, x 3) for xy, : x p ( x y ) ( x y, x y ) ( x, x y ) ( y, x y ) p( x) p( x y) p( y) p( x y) p( x y)[ p( x) p( y)] So p( x y) [ p( x) p( y)], for xy, MAIN RESULTS Definition The function, :X X, where X is a vectorial space, it is called the asymmetric quasi inner product if: a) ( xx, ), x X b) ( x, y) ( x, y), ( x, y) X and ( x, y) ( x, y), ( x, y) X and c) ( x x, y) ( x, y) ( x y), x, x, y X d) ( x, y) k( x, x)( y, y), for k 1 Definition 3 The function p: X it is called the asymmetric quasi norm function if: a) px ( ), x X b) p( x) p( x), x X and c) p( ) k p( x) p( y), ( x, y) X and k 1 Proposition If ( xy, ) is the asymmetric quasi inner product function on X, then the function p: X such that p( x) ( x, x) is an asymmetric quasi norm function Proof 1) We have p( x) ( x, x), x X ) We have Therefore, for p( x) ( x, x) ( x, x) p( x) ( x, x) p( x)

7 Asymmetric Inner Product and 67 3) We have p ( x y ) ( x y, x y ) ( x, x y) ( y, x y) ( x, x y) ( y, x y) k ( x, x)( x y, x y) k ( y, y)( x y, x y) k p( x) p( x y) k p( y) p( x y) k p( x) p( y) p( x y) From where: p( x y) k p( x) p( y) and if we denote k 1 have: p( x y) k[ p( x) p( y)] 3 CONCLUSIONS An asymmetric quasi norm function can be obtained by an asymmetric inner product function, and the link between them is the function: p: X, so that p( x) ( x, x) References [1] G Lumer Semi- inner products spacestrans Am MathSoc 1 (1961), 9-43 [] R GilesClasses of semi inner product spaces TransAm MathSoc19 (1967), [3] L Tashim On denseness on asymetric Metric Spaces Mathematics Aeterna, Vol1 [4] S Cobzas Functional on Asymmetric Normed Spaces Springer Fronties in Mathematics, 13 ISBN ) Faculty of Natural Sciences, Department of Mathematics, Aleksandër Xhuvani University, Elbasan, Albania address: stelaceno85@yahoocom ) Faculty of Natural Sciences, Department of Mathematics, Aleksandër Xhuvani University, Elbasan, Albania address: gladitigno@yahoocom