Fuzzy n-normed Space and Fuzzy n-inner Product Space

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1 Global Joural o Pure ad Applied Matheatics. ISSN Volue 3, Nuber 9 (07), pp Research Idia Publicatios Fuzzy -Nored Space ad Fuzzy -Ier Product Space Mashadi ad Abdul Hadi Departet o Matheatics Faculty o Matheatics ad Natural Scieces, Uiversity o Riau Pekabaru, Idoesia. Abstract I this paper we costruct a ew cocept o uzzy -ored space ad uzzy -ier product space. First, we show that every -ored space ca be costructed as a uzzy -ored space. The we show that every -ored space is a uzzy -ored space ad every uzzy -ored space ca also be reduced to a uzzy -ored space. Furtherore, or every -ier product space ca be costructed a uzzy -ier product space. Fially we show that every -ier product space is a uzzy -ier product space ad every uzzy -ier product space is a uzzy -ored space. Keywords: Fuzzy -ored space, uzzy -ored space, uzzy -ier product space. INTRODUCTION Fuzzy set theory was irst itroduced by L. Zadeh o 965. This theory cotiues to evolve i various disciplies. I atheatics, especially atheatical aalysis, uzzy cocept has bee developed i uzzy -ored space ad uzzy -ier product. Util ow, it has bee widely oud outcoes related to uzzy ored space ad uzzy ier product space. It has bee prove i [6] that every -ored space is a ored space ad i [3] ad [] has also bee deostrated that the ored space ad uzzy ored space is equivalet so that every -ored space is certaily a uzzy ored space. O the other had, i [5] it has also bee deostrated that the -ored spaces ad uzzy -ored spaces are also equivalet. However, uexplored i geeral how are the relatioship betwee -ored space with uzzy

2 4796 Mashadi ad Abdul Hadi -ored space ad uzzy -ored space with uzzy ored space. Moreover, [7] has proved that every -ier product space is a -ored space. Ad [] has also proved that the ier product space ad uzzy ier product space are equivalet. Moreover, it also has bee prove i [5] that every -ier product space is a space uzzy -ier product. However, uexplored i geeral how are the relatioship betwee the -ier product space ad uzzy - ier product space ad uzzy -ored space. I this paper, irst we give a deiitio o uzzy -ored ad uzzy -ier product that are dieret ro deiitio o uzzy -ored i [] ad uzzy -ier product i [7] which is also dieret ro the deiitio put orward by other authors. The soe theores are give to shows the relatioship aog -ored space, uzzy -ored space, uzzy ored space, -ier product ad uzzy -ier product space. For ore details, oe ca see a chart i the Figure. (Blue arrows show the ew theore i this paper) NS [5] NS NS [6] ( )NS [,3] FNS [3] [5] FNS [5] [7] FNS F( )NS IPS [] [9,] IPS [6] [5] IPS FIPS [5] FIPS FIPS Notes: NS: -Nor Space; FNS: Fuzzy -Nored Space; IPS: -Ier Product Space; FIPS : Fuzzy -Ier Product Space. Figure. The relatioship aog -ored space, uzzy -ored space, -ier product space ad uzzy -ier product space. DISCUSSION Let X be ay oepty set. The a uzzy set A i X is characterized by a ebership uctio : X [0,]. The A ca be writte as A A {( x, ( x )) x X,0 ( x ) }. A A

3 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 4797 Deiitio o uzzy poit has bee widely discussed by various authors, aog others are as ollows [9-3 ad 5]: Deiitio A Fuzzy poit P x i X is a uzzy set whose ebership uctio is i y x P x ( y) 0 i y x or all y X, where 0. We deote uzzy poits as x or ( x, ). Deiitio Two uzzy poits x ad y is said equal i x y ad where, (0,]. Deiitio 3 Let x be a uzzy poit ad A be a uzzy set i X. The x is said belog to A or x A i ( x ). A Let X be a vector space over a ield K ad A be a uzzy set i X. The A is said to be a uzzy subspace i X i or all x, y X ad K satisy (i) ( x y) i{ ( x), ( y)} A A A (ii) ( x) ( x). A A Fuzzy ored space is deied i various ways. The deiitio based o a istituiistic approach is dieret ro the uzzy poit approach. The ollowig are give the deiitio o uzzy ored space based o the uzzy poit approach as discussed o [5], ad deiitio o uzzy ored used i this paper is as give by [] ad [5]. Deiitio 4 Let X be a vector space over a ield K, be a uctio that associates each poit x uber so that (FN) x 0 i oly i x 0. (FN) x x or all K. i A, (0,] ad let : A [0, ) to oegative real

4 4798 Mashadi ad Abdul Hadi (FN3) x y x y. (FN4) I 0, the x x ad there exists 0, such that li x x. The is called a uzzy or o X ad a pair ( X, ) is called a uzzy ored space. Deiitio 5 Let N da X be a vector space over R with di( X). Deie uctio,, : X X X R with ( x, x,..., x) x, x,..., x. ties Fuctio,, is called -or o X i or every x, x,..., x, y, z X satisyig the coditios: (N) x, x,..., x 0 i oly i x, x,..., x liearly depedet. (N) x, x,..., x ivariat uder perutatio. (N3) x, x,..., x x, x,..., x or ay R. (N4) x, x,..., x, y z x, x,..., x, y x, x,..., x, z. A pair ( X;,, ) is called -ored space. Fro deiitio o -or ad uzzy or o X above, based o the -ored costructio o [6], the author costructs a ew uctio to produce a ew space that called uzzy -ored as ollows. Deiitio 6 Let N ad X be a vector space over R.,, : A A A 0, ties Deie uctio where ( x, x,..., x ) x, x,..., x () ( ) () ( ) () ( ) () ( ) 0, ad x A or i,,.... Fuctio,, is called Fuzzy with -ored i satisy the coditios : x, x,..., x 0 i oly i () ( ) (FN) () ( ) () ( ) x, x,..., x liearly depedet.

5 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 4799 x, x,..., x ivariat uder perutatio. () ( ) (FN) () ( ) () ( ) () ( ) (FN3) () ( ) () ( ) x, x,..., rx r x, x,..., x or r R. (FN4) x x ( ) y z x x ( ) y x x ( ) z ( ) ( ) ( ),...,,,...,,,...,,. (FN5) I 0 or,,..., i i the x, x..., x x, x,..., x () ( ) () ( ) () ( ) ad there exist i ad N such that 0 or,,..., li x, x..., x x, x..., x. () ( ) () ( ) () ( ) () ( ) A pair ( X,, ) is called Fuzzy -Nored Space. I [3], [] ad [0] it is proved that every ored space is uzzy ored space. Coversely is also true. I other side, i [5] it is also etioed that -ored space ad uzzy -ored space is equivalet. The ollowig is give a theore that esures that ay every -ored spaces is uzzy -ored space. Theore 7 Let N, X be a vector space over R ad di( X). Suppose ( X;,..., ) is -ored space. Deie where x, x,..., x x, x,..., x () ( ) () ( ) () ( ) ax,,,. The ( X,..., ) is uzzy -ored space. Proo. Suppose x, y, z A, (0,] or i,,..., ad r R. The (FN) () ( ) x, x,..., x 0 x, x,..., x 0 () ( ) () ( ) () ( ) x, x,..., x 0 () ( ) x, x,..., x are liearly depedet. (FN) Sice () ( ) x, x,..., x is ivariat uder perutatio, the

6 4800 Mashadi ad Abdul Hadi () ( ) x, x (),..., x ( ) is ivariat uder perutatio. (FN3) () ( ) () ( x ), x (),..., rx ( ) x, x,..., rx r x, x,..., x r x, x,..., x () ( ) () ( ) r x, x,..., x () ( ) ( ) ( ) ( ) ( ) ( ) (FN4) ( ) ( ) x,..., x, y z x,..., x,( y z) with ax{, ) x,..., x, y z ad ax{,...,, } ( ) ( ) x,..., x, y x,..., x, z ( ) ( ) x,..., x, y x,..., x, z ( ) ( ) x,..., x, y x,..., x, z ( ) ( ) ( ),..., x x ( ), z ( ) x (,..., ) x (, y) where ax{,...,, } ad ax{,...,, } ( ) ( ) (FN5) Suppose ax { } ad i{,..., } i ax { }. i{,..., } I i i, the, such that 0 or,,...,. The x, x..., x x, x,..., x x, x,..., x () ( ) () ( ) ( ) ( ) () ( ) x, x,..., x. () ( ) Further, suppose Hi { r 0 r } or i,,...,. Sice Hi is bouded, the there is a sequece di Hi or i,,..., that coverges. Costruct () sequece i where or i,,...,. The sequece

7 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 480 coverges to or i,,...,. I sequece coverges to such that ad () ax { i } i{,..., } the ax { }, i{,..., } li x, x..., x li x, x,..., x () ( ) () ( ) ( ) ( ) x, x,..., x () ( ) x, x..., x. () ( ) ( ) ( ) ( ) I [6], it is described that every -ored space is or space. The ollowig is give a theore which states that uzzy -ored space is uzzy ored space. Theore 8 Let ( X,,..., ) be a uzzy -ored space or all. Deie x x, a,..., a x, a ( ) ( ) ( ) ( ) ( ) where a,..., a ( ) ored space. are liearly idepedet vectors. The ( X,. ) is uzzy Proo. (FN) I x 0 the obviously x 0. I x 0 the ( ) ( ) x, a,..., a x, a 0. Sice x, a ( ) 0, the x, a ( ) ( ) ( ) ( ) ( ) liearly ( ) ( ) depedet. But, sice a ( ) liearly idepedet, the x t a ( ) or soe t R. x, a,..., a 0, ( ) Sice ( ) the ( ) I other side, sice a,..., a ( ) ( ) ( ) ( ) ( ) ta, a,..., a 0 ( ) ( ) ( ) ( ) t a, a,..., a 0. ( ) ( ) ( ) ( ) are liearly idepedet vectors. The a, a,..., a 0 so ust ecessarily t 0 or t 0. Cosequetly, x t a 0a 0. ( ) ( ) ( ) ( ) x x, a,..., a x, a ( ) ( ) (FN) ( ) ( )

8 480 Mashadi ad Abdul Hadi x a,,..., a x a, ( ) ( ) ( ) ( ) ( ) x a,,..., a x a, x. ( ) ( ) ( ) ( ) ( ) x y x y, a,..., a x y, a ( ) ( ) (FN3) ( ) ( ) x, a,..., a y a,,..., a x a, y a, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x, a,..., a x a, y, a,..., a y a, x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y (FN4) I 0 the use (FN4), So that x, a,..., a x, a,..., a ad x, a x, a. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x x, a,..., a x, a x, a,..., a x, a x. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Further, there exist i ad N 0 or,,..., so usig (FN5), li x, a,..., a x, a,..., a ad li x, a x, a. So that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) li x li x, a,..., a x, a ( ) ( ) ( ) ( ) li x, a,..., a li x, a ( ) ( ) ( ) ( ) x, a,..., a x, a ( ) ( ) ( ) ( ) x. I [6], it be proved that every -ored space is (-) ored space. The ollowig, we will proo a theore which states that every uzzy -ored space is uzzy (-) ored space.

9 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 4803 Theore 9 Let ( X,,..., ) be uzzy -ored space or all. Deie () ( ) () ( ) ( i), (),..., ( ) ax, (),..., ( ), ( i) i{,..., } x x x x x x a ( ) where a,..., a ( ) (-)-ored space. Proo. are liearly idepedet vectors. The ( X,.,...,. ) is uzzy () ( ) () ( ) ( i) I x, x (),..., x ( ) are liearly depedet, the x, x (),..., x ( ), a ( i) or () ( ) ( i) i,..., are liearly depedet too so that x, x (),..., x ( ), a ( i) 0 () ( ) i,...,. As a result, x, x (),..., x ( ) 0. Coversely, i () ( ) () ( ) ( i) x, x (),..., x ( ) 0, the x, x (),..., x ( ), a ( i) 0 or i,...,. () ( ) ( i) The x, x (),..., x ( ), a ( i) a,..., a ( ) ( ) or or i,..., are liearly depedet. Sice are liearly idepedet vectors, the we ust have () ( ) x, x (),..., x ( ) liearly depedet. () ( ) ( i) () Sice x, x (),..., x ( ), a ( i) x, x,..., x () ( ) () ( ) is ivariat uder perutatios, the is ivariat uder perutatios too. (3) () ( ) () ( ) ( ) () ( ) () (,,..., ax,,..., ), ( i x x rx ) x x rx a i i{,..., } () ( ) ( i) ax r x, x (),..., x ( ), a ( i) i{,..., } () ( i ( ) r ax x x, (),..., x ( a) i, ( ) i{,..., } () ( ) () ( ) r x, x,..., x or all r R. (4) () ( ) () ( ) ( ) x, x (),..., x ( ), y z ax x, x (),..., x ( ), y z, a ( i ) i i{,..., } () ( ) ( i) () ( ) ( i) ax x, x (),..., x ( ), y, a ( i) x, x (),..., x ( ), z, a ( i) i{,..., } () ( ) ( i) () ( ) ( i) ax x, x (),..., x ( ), y, a ( i) ax x, x (),..., x ( ), z, a ( i) i{,..., } i{,..., } () ( ) () ( ) () ( ) () ( ) x, x,..., x, y x, x,..., x, z

10 4804 Mashadi ad Abdul Hadi (5) I ad there exist 0 or,,..., i i the () ( ) () ( ) ( i) (,)..., ( ) ( ) ax, (,..., ) ( ), i ( ) ( ) i{,..., } x x x x x x a ax x, x,..., x, a () ( ) ( i) i{,..., } ( i ) () ( ) x, x,..., x 0 or i,,..., ad ad N such that () ( ) () ( ) ( i) li x, x ()..., x ( ) li ax x, x ()..., x ( ), a ( i) {,..., } i ax li x, x..., x, a () ( ) ( i) () ( ) ( i) i{,..., } ax x, x,..., x, a () ( ) ( i) () ( ) ( i) i{,..., } x, x,..., x () ( ) () ( ) 0 or i,,..., Corollary 0 Every uzzy -ored space is a uzzy r-ored space or r,...,. I particular, every uzzy -ored space is a uzzy ored space. Proo. This is a direct result o Theore 9. Corollary Every -ored space is a uzzy ored space. Proo. Apply Theore 7 da 8. Whereas the relatioship betwee the ored space ad the uzzy ored space ad vice versa has bee discussed i [5] ad [6]. Theore Let ( X,. ) be a ored space. Deie x x or all x X where (0,]. The ( X,. ) is a uzzy ored space.

11 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 4805 Theore 3 Let ( X,. ) be uzzy ored space. Deie x x ( x,) or all x X. The ( X,. ) is a or space. Corollary 4 Every uzzy -ored space is a or space. Proo. Apply Corollary 0 da Theore 3. The ollowig is give the deiitio o -ier product that is dieret ro the deiitio i [7] ad []. Deiitio 5 Let N ad, X be a vector space over R ad di( X). Fuctio,,, : X X X R is called -ier product o ties X i or every x, x,..., x, y, z X satisy the coditios : (IP) x, x x,..., x 0 ad x, x x,..., x 0 i oly i x, x,..., x liearly depedet. (IP) x, y x,..., x y, x x,..., x. (IP3) x, y x,..., x ivariat uder perutatio x,..., x. (IP4) 3 x, x x,..., x x, x x, x,..., x. (IP5) x, x x,..., x x, x x,..., x or R. (IP6) y z x x x y x x x z x x x,,...,,,...,,,...,. A pair ( X,,,, ) is called -ier product space. Deiitio 6 Let X be a vector space over K C atau K R. Deie uctio, : A A K with x, y x, y ( ) or every x, y A, where i{, }, ad, (0,]. For every x, y, z A, i{,, } ad,, (0,]. Fuctio, is called uzzy ier product i it satisies the coditios:

12 4806 Mashadi ad Abdul Hadi (FIP) x, x ( ) 0 ad x, x ( ) 0 i oly i x 0. (FIP) x, y ( ) y, x ( ). (FIP3) rx, y ( ) r x, y ( ) or r K. (FIP4) x y, z ( ) x, z ( ) y, z ( ). (FIP5) I 0 the x, x ( ) x, x ( ), ad there exist 0 so that li x, x ( ) x, x ( ). A pair ( X,, ) is called uzzy ier product space. Fro deiitio o -ier product ad uzzy ier product o X above, ca be costructed a ew uctio called uzzy -ier product as ollows. Deiitio 7 Let N ad, X be a vector space over R ad di( X). Deie uctio with where ~ ~ ~,,, : A A A 0, ties x, x x,..., x x, x x,..., x ( ) () ( ) () ( ) () ( ) i { }, i(,..., } 0, ad ( x ) A or i,,.... Fuctio,,, is called uzzy -ier product i it satisies the coditios : (FIP) () ( ) () ( ) x x x x x x x P X,,..., ( ) 0 or,,..., ( ) ad depedet. () ( ) x, x x,..., x ( ) 0 i oly i () ( ) x, x,..., x are liearly (FIP) () ( ) () ( ) x, y x,..., x ( ) y, x x,..., x ( ) where i {, }. i(,..., } (FIP3) i {, }. i(,..., } () ( ) x, y x,..., x ( ) ivariat uder perutatio where (FIP4) () ( ) () () (3) ( ) x, x x,..., x ( ) x, x x, x,..., x ( ).

13 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 4807 (FIP5) (FIP6) () ( ) () ( ) rx x x x r x x x x r R,,..., ( ),,..., ( ) or. () ( ) () ( ) () ( ) x z, y x,..., x ( ) x, y x,..., x ( ) z, y x,..., x ( ). where i {,, }. i(,..., } (FIP7) I 0 the ad there exist 0 so that () ( ) () ( ) x, x x,..., x ( ) x, x x,..., x ( ) () ( ) () ( ) li x, x x,..., x ( ) x, x x,..., x ( ). A pair ( X,,,, ) is called uzzy -ier product space. I [] it is proved that every ier product space is uzzy ier product space. Coversely is also true. The ollowig is provided theore that esure that every -ier product space is uzzy -ier product space. Theore 8 Let ( X,,,, ) be a -ier product space. Deie () () ( ) () ( ) x, x x (),, x x, x x,, x ( ) x, x x,, x. The ( X;,,, ) is a uzzy -ier product space. Proo. (FIP) () ( ) () ( ) x, x x,, x ( ) x, x x,, x 0. () ( ) () ( ) x, x x,, x ( ) 0 x, x x,, x 0 () ( ) x, x x,, x 0 () ( ) x x,, x liearly depedet. (FIP) x, y x,..., x ( ) x, y x,..., x () ( ) () ( )

14 4808 Mashadi ad Abdul Hadi (FIP3) Applyig (IP3), that is () ( ) x,..., x. The perutatio where y, x x,..., x () ( ) y x x x () ( ),,..., ( ). () ( ) x, y x,..., x is ivariat uder perutatio () ( ) () ( ),,..., x y x x ( ) x, y x,..., x ivariat uder i {, }. i(,..., } (FIP4) (FIP5) (FIP6) x, x x,..., x ( ) x, x x,..., x () ( ) () ( ) x, x x, x,..., x () () (3) ( ) () () (3) ( ) x, x x, x,..., x ( ). rx, x x,..., x ( ) rx, x x,..., x () ( ) () ( ) () ( ) r x, x x,..., x () ( ) r x, x x,..., x. () ( ) r x x x x r x z, y x,..., x ( ) x z, y x,..., x () ( ) () ( ),,..., ( ) or all. x, y x,..., x z y x, x,..., x, y x,..., x z, y x,..., x () ( ) () ( ) () ( ) () ( ) x, y x,..., x ( ) z, y x,..., x ( ). () ( ) () ( ) where i {,, }. i(,..., } (FIP7) I 0 the. We get

15 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 4809 x, x x,..., x x, x x,..., x () ( ) () ( ) () ( ) () ( ) x, x x,..., x ( ) x, x x,..., x ( ) ad there exists 0 with coverges to. So that li x, x x,..., x ( ) li x, x x,..., x () ( ) () ( ) x, x x,..., x () ( ) () ( ) x x x x,,..., ( ). I [] it is proved that every uzzy ier product space is uzzy ored space. However, coversely is ot true. For exaple, oe ca see i [9]. I other side, i [5] it proved that uzzy -ier product space is uzzy -ored space, but or geeral case has ot yet provided. The ollowig is give a theore which esures that every uzzy -ier product space is a uzzy -ored space. Theore 9 Let ( X,,,, ) be a uzzy -ier product space or. Deie x, x,..., x x, x x,, x () ( ) () ( ) () ( ) () ( ) with 0, ad x A or i,,.... the ( X;,..., ) is a uzzy -ored space. () ( ) () ( ) Proo.(FN) x, x (),..., x ( ) 0 x, x x (),, x ( ) 0 () ( ) (FN) () ( ) () ( ) x, x,..., x liearly depedet. x, x,..., x ivariat uder perutatio. (FN3) rx, x,..., x rx, rx x,, x () ( ) () ( ) () ( ) () ( )

16 480 Mashadi ad Abdul Hadi r x, rx x,, x () ( ) ( ) ( ) r rx, x x,, x () ( ) ( ) ( ) r x, x x,, x () ( ) ( ) ( ) r x, x x,, x () ( ) ( ) ( ) () ( ) r x, x ( ),..., ( ) x or all r R. (FN4) ( ) ( ) ( ) x,..., x, y z y z, x,..., x ( ) y z, y z x,, x ( ) ( ) y, y z x,, x z, y z x,, x ( ) ( ) ( ) ( ) y z, y x,, x y z, z x,, x ( ) ( ) ( ) ( ) ( ) ( ) ( ) y, y x,, x ( ) z, y x,, x ( ) y, z x,, x ( ) z, z x,, x ( ) ( ) y, y x,, x y, z x,, x z, z x,, x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y, x,..., x y, x,..., x z, x,..., x z, x,..., ( ) y, x,..., x z, x,..., x so that ( ) ( ) ( ) ( ) x,..., x, y x,..., x, z. ( ) ( ) ( ) ( ) (FN5) I ( ) ( ) ( ) ( ) ( ) ( ) x,..., x, y z x,..., x, y x,..., x, z. 0 or,,..., i i the x x, x..., x x, x x,, x () ( ) () ( ) ( ) ( ) ( ) ( ) x, x x,, x () ( ) x, x,..., x. () ( )

17 Fuzzy -Nored Space ad Fuzzy -Ier Product Space 48 ad there exist i ad N so that 0 or,,..., li x, x..., x li x, x x..., x () ( ) () ( ) ( ) ( ) ( ) ( ) li x, x x..., x () ( ) ( ) ( ) x, x x,, x () ( ) ( ) ( ) x, x..., x. () ( ) ( ) ( ) ( ) Corollary 0 Every -ier product space is uzzy -ored space. Proo. Apply Theore 8 ad 9. REFERENCES [] Bill Jacob., 989, Liear Algebra, W.H. Freea ad Copay, New York. [] Erwi, K., 978, Itroductory Fuctioal Aalysis with Applicatios, Joh Wiley ad Sos. Ic. [3] Guawa.H., ad Mashadi., 00, "O iite-diesioal -ored spaces, Soochow J. Math, 7, pp [4] Guawa, H., ad Mashadi., 00, O -ored spaces, It. J. Math. Math. Sci., 7,pp [5] Guawa, H., 00, Ier Products O -Ier Product Spaces, Soochow Joural O Matheatics., 8, pp [6] Guawa, H., 00, O -ier products, -ors ad the Cauchy-Schwarz iequality, Sci. Math. Jp., 55, pp [7] Kider, J. R., ad Sabre, R.I., 00, Fuzzy Hilbert Spaces, Eg. Tech. J. 8, pp [8] Kider, J. R., 0, O Fuzzy Nored Spaces, J. Egieerig ad Techology, 9, pp [9] Kider, J. R., 0, Copletio o uzzy ored space, Eg. Tech. J. 9, pp [0] Kider, J. R., 004, Fuzzy Locally Covex, Algebra, Ph.D. Thesis. Techology Uiversity, Baghdad.

18 48 Mashadi ad Abdul Hadi [] Kider, J. R, 0, New Fuzzy Nored Spaces, J. Baghdad Sci., 9, pp [] Mashadi., ad Sri Geawati., 05, Soe Alterative Cocepts or Fuzzy Nored Space ad Fuzzy -Nored Space, JP Joural o Matheatical Scieces, 4, pp [3] Misiak, A., 989, -ier product spaces, Math. Nachr., 40, PP [4] Narayaa., ad Vijayabalaji, S., 005, Fuzzy -ored liear space, It. J. Math. Math. Sci., 4, PP [5] Parijat Siha, O Fuzzy -Ier Product Spaces, Iteratioal Joural o Fuzzy Matheatics ad Systes, 3 (03), [6] S. Vijayabalaji ad Natesa Thillaigovida, Fuzzy -Ier Product Space, Korea Matheatical Society, 43 (007), [7] Zadeh,L.A., 965, Fuzzy Set, Ioratio ad Cotrol, 8, pp