Some Fundamental Properties of Fuzzy Linear Relations between Vector Spaces

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1 Filomat 30:1 2016, DOI /FIL N Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Some Fundamental Properties of Fuzzy Linear Relations between Vector Spaces Sorin Nădăban a a Aurel Vlaicu University of Arad, Department of Mathematics and Computer Science, Str Elena Drăgoi 2, RO Arad, Romania Abstract This paper aims at studying the fundamental properties of fuzzy linear relations between vector spaces The sum of two fuzzy relations and the scalar multiplication are defined, in a natural way, and some properties of this operations are established Fuzzy linear relations are investigated and among the results obtained, there should be underlined a characterization of fuzzy linear relations and the fact that the inverse of a fuzzy linear relation is also a fuzzy linear relation Moreover, the paper shows that the composition of two fuzzy linear relations is a fuzzy linear relation as well Finally, the article highlights that the family of all fuzzy linear relations is closed under addition and it is closed under scalar multiplication 1 Introduction Linear relations were introduced by R Arens [1] in 1961 Since then, they have been a preoccupation for many mathematicians It was only in 1998 that the theory of linear relations was systematized in a beautiful monograph by R Cross [8] The subject is not closed though Thus, in 2003, A Száz [15] investigated the possibility of extending of linear relation The development of spectral theory for linear relations was the aims of recent papers: in 2002 AG Baskakov and KI Chernyshov [2], in 2007 AG Baskakov and AS Zagorskii [3], in 2012 D Gheorghe and F-H Vasilescu [11] It has to be mentioned that D Gheorghe and F-H Vasilescu study in paper [10] linear maps defined between spaces of the form X/X 0, where X is a vector space and X 0 is a vector subspace of X The motivation of this approach comes from the theory of linear relations On the other hand, the concept of fuzzy set introduced by L Zadeh [19] in 1965, represented a natural frame for generalizing many of the concepts of mathematics The introduction in 1977 by AK Katsaras and DB Liu [13] of the concept of fuzzy topological vector space resulted in what we can call today a new mathematical field Fuzzy Functional Analysis The present paper is based on results refereing to fuzzy linear subspaces obtained by AK Katsaras and DB Liu In 1985, NS Papageorgiou [14] introduced the notion of fuzzy multifunction and started the study of linear fuzzy multifunction The investigation of fuzzy multifunctions was continued, through a series of papers by E Tsiporkova, B De Baets, E Kerre [17], [18] and I Beg [4], [5], [6], [7] A brief survey, concerning weakly linear systems of fuzzy relation inequalities and their applications in fuzzy automata, the study of simulation and in the social network analysis, was made in paper [12] by J Ignjatović and M Ćirić There are also some recent papers in this 2010 Mathematics Subject Classification Primary 47A06 ; Secondary 03E72, 47S40 Keywords Fuzzy relation; fuzzy linear relation; fuzzy linear operator; fuzzy multivalued functions Received: 12 April 2014; Accepted: 12 August 2015 Communicated by Hari M Srivastava address: snadaban@gmailcom Sorin Nădăban

2 S Nădăban / Filomat 30:1 2016, field see [9] In [16], B Šešelja, A Tepavčevic and M Udovičic are dealing with fuzzy posets and their fuzzy substructures The aim of this paper is to study the fundamental properties of fuzzy linear relations between vector spaces Some results in the present paper may look similar, at the first sight, to I Beg s results, but they are not The differences between them are significant Along this paper, we have identified fuzzy sets with their membership functions, while, I Beg identified the fuzzy set with their support Thus, in this paper, all equalities and inclusions are between fuzzy sets as compared to some of I Beg s results where the obtained equalities or inclusions are only between the support sets of some fuzzy sets In this paper the sum of two fuzzy relations and the scalar multiplication are defined, in a natural way, and some properties of this operations are established Fuzzy linear relations are investigated and among the results obtained, there should be underlined a characterization of fuzzy linear relations and the fact that the inverse of a fuzzy linear relation is also a fuzzy linear relation Moreover, the paper shows that the composition of two fuzzy linear relations is a fuzzy linear relation as well Finally, the article highlights that the family of all fuzzy linear relations is closed under addition and it is closed under scalar multiplication 2 Fuzzy Linear Subspaces Let X be a nonempty set A fuzzy set in X see [19] is a function µ : X [0, 1] We denote by F X the family of all fuzzy sets in X The symbols and are used for the supremum and infimum of a family of fuzzy sets We write µ 1 µ 2 if µ 1 x µ 2 x, x X Let f : X Y If µ F Y, then f 1 µ : µ f If ρ F X, then f ρ F Y is defined by f ρy : ρx : x f 1 y} if f 1 y 0 otherwise If µ F X, the support of µ is supp µ : x X : µx > 0} Let X be a vector space over K where K is R or C If µ 1, µ 2,, µ n are fuzzy sets in X, then µ µ 1 µ 2 µ n is a fuzzy set in X n defined by see [13] µx 1, x 2,, x n µ 1 x 1 µ 2 x 2 µ n x n Let f : X n X, f x 1, x 2,, x n n x k The fuzzy set f µ is called the sum of fuzzy sets µ 1, µ 2,, µ n and it is denoted by µ 1 + µ µ n see [13] In fact k1 µ 1 + µ µ n x µ 1 x 1 µ 2 x 2 µ n x n : x n x k k1 If µ F X and K, then the fuzzy set µ is the image of µ under the map : X X, x x Thus see [13] µ x if 0 µx 0 if 0, x 0 µy : y X} if 0, x 0 Proposition 21 Let µ 1, µ 2 F X and K Then µ 1 + µ 2 µ 1 + µ 2 Proof It is obvious Definition 22 [13] Let X be a vector space over K µ F X is called fuzzy linear subspace of X if 1 µ + µ µ; 2 µ µ, K We de note by FLSX the family of all fuzzy linear subspace of X

3 S Nădăban / Filomat 30:1 2016, Proposition 23 [13] Let X be a vector space over K and µ F X The following statements are equivalent: 1 µ FLSX; 2 α, β K, we have αµ + βµ µ; 3 x, y X, α, β K, we have µαx + βy µx µy Proposition 24 [13] If µ, ρ FLSX and K, then: 1 µ + ρ FLSX; 2 µ FLSX; 3 µx µ0, x X 3 Fuzzy Relations In this section we consider X, Y be two nonempty sets A fuzzy relation T or fuzzy multifunction, or fuzzy multivalued function between X and Y is a fuzzy set in X Y, ie a mapping T : X Y [0, 1] For x X, we denote by T x the fuzzy set in Y defined by: T x : Y [0, 1], T x y Tx, y Thus a fuzzy relation T can be seen as a mapping X x T x F Y see [14] We denote by FRX, Y the family of all fuzzy relations between X and Y If X Y, then we set FRX FRX, Y The domain DT of T is a fuzzy set in X defined by DTx : sup Tx, y see [18] We note that supp DT x X : T x } x X : y Y such that Tx, y > 0} If for all x supp DT there exists an unique y Y such that Tx, y > 0, then T is called fuzzy function or single-valued fuzzy function In this case, we denote this unique y by Tx If µ F X, then Tµ F Y is defined by Tµy : sup[tx, y µx] see [4] In particular, the range RT of T is a fuzzy set in Y defined by RTy : sup Tx, y see [18] Let T FRX, Y, S FRY, Z The composition S T FRX, Z or simply ST is defined by see [19]: S Tx, z : sup[tx, y Sy, z] Proposition 31 Let T FRX, Y, S FRY, Z Then S T x ST x, x X Proof It is obvious Proposition 32 The operation is associative Proof It is obvious Definition 33 Let T FRX, Y If E X, then the fuzzy relation T E defined by T E : E Y [0, 1], T E x, y Tx, y is called the restriction of T to E Moreover, the fuzzy relation T is called an extension to X of a fuzzy relation S FRE, Y if S T E The inverse or reverse relation T 1 of a fuzzy relation T FRX, Y is a fuzzy set in Y X defined by T 1 y, x Tx, y It is obvious that RT DT 1 and RT 1 DT We remark that, for µ F Y, we have T 1 µx sup[t 1 y, x µx] sup[tx, y µx] This type of inverse is usually called lower inverse see [4] T FRX, Y is called surjective if supp RT Y A fuzzy relation T FRX, Y is called injective if, for x 1 x 2, we have T x1 T x2 This means that, for x 1 x 2, we have Tx 1, y Tx 2, y 0, y Y A fuzzy relation I FRX is called identity relation if Ix, y 0 for all x, y X, x y If we denote by E the support of DI, the identity relation will be denoted I E If µ F X, by identity relation on µ we understand Isupp µx, x µx

4 Proposition 34 Let T FRX, Y Then 1 supp DT X if and only if I X T 1 T; 2 T is injective if and only if T 1 T Isupp DT; 3 T is a fuzzy function if and only if TT 1 Isupp RT; 4 T is injective if and only if T 1 is a fuzzy function Proof First we note that, for x 1, x 2 X, we have 1 For x X, we have S Nădăban / Filomat 30:1 2016, T 1 Tx 1, x 2 sup[tx 1, y T 1 y, x 2 ] sup[tx 1, y Tx 2, y] T 1 Tx, x sup[tx, y Tx, y] sup Tx, y DTx supp DT X DTx > 0, x X T 1 Tx, x > 0, x X T 1 T I X 2 For x 1 x 2 we have Tx 1, y Tx 2, y 0, y Y T 1 Tx 1, x 2 0 For x X we have T 1 Tx, x sup Tx, y DTx Thus T 1 T Isupp DT Let x 1 x 2 As T 1 T Isupp DT, we have that T 1 Tx 1, x 2 0 Thus, for all y Y, we have Tx 1, y Tx 2, y 0 Hence T x1 T x2 3 TT 1 y 1, y 2 sup[t 1 y 1, x Tx, y 2 ] sup[tx, y 1 Tx, y 2 ] For y 1 y 2, we have that Tx, y 1 0 or Tx, y 2 0, for all x X Hence TT 1 y 1, y 2 0 On the other hand TT 1 y, y sup Tx, y RTy Thus TT 1 Isupp RT We suppose that T is not a fuzzy function Then x supp DT, y 1, y 2 Y, y 1 y 2 such that Tx, y 1 > 0, Tx, y 2 > 0 TT 1 y 1, y 2 > 0, contradiction 4 T is injective T 1 T Isupp DT T 1 T Isupp RT 1 T 1 is a fuzzy function 4 Fuzzy Relations between Vector Spaces In this section we consider X, Y be two vector spaces Definition 41 Let T, S FRX, Y and K We define the sum T + S FRX, Y and scalar multiplication T FRX, Y by T + Sx, y sup [Tx, y 1 Sx, y 2 ] and Tx, y T x, y DTx if y 0, if 0 ; 0Tx, y For K we define the fuzzy function Y FRY by Y x, y 1 if y x 0 if y x In fact this is an ordinary function, precisely the function Y x x Y Proposition 42 Let T FRX, Y and K Then T Y T

5 Proof Case 1 0 S Nădăban / Filomat 30:1 2016, Y Tx, z sup[tx, y Y y, z] T x, z z Y, z T x, z Tx, z Case 2 0 Y Tx, z sup[tx, y Y y, z] sup Tx, y if z 0 0 if z 0 DTx if z 0 0 if z 0 Tx, z Proposition 43 Let T, S FRX, Y and K Then 1 DT + S DT DS; 2 DT DT Proof 1 DT + Sx supt + Sx, y sup sup [Tx, y 1 Sx, y 2 ] sup [Tx, y 1 Sx, y 2 ] y 1,y 2 Y 2 Case 1 0 Case 2 0 sup y 1 Y Tx, y 1 sup Sx, y 2 DTx DSx [DT DS]x y 2 Y DTx sup Tx, y sup T x, y sup Tx, y DTx y Y D0Tx sup0tx, y 0Tx, 0 DTx Proposition 44 Let T FRX, Y and K Let sup T : suptx, y : x X, y Y} Then sup T sup T Proof Case 1 0 sup T : suptx, y : x X, y Y} supt x, y : x X, y Y} Case 2 0 suptx, y : x X, y Y} sup T sup T : sup0tx, y : x X, y Y} sup0tx, 0 sup DTx sup sup Tx, y sup T Theorem 45 Let T, S, R FRX, Y and α, β K Then 1 T + S + R T + S + R; 2 T + S S + T; 3 T + 0 T, where 0 FRX, Y is defined by 0x, y 4 αt + S αt + αs; 5 αβt αβt; 6 1T T 1 if y 0 ;

6 S Nădăban / Filomat 30:1 2016, Proof 1 Let x, y X Y Then 2 and 3 are obvious 4 Case 1 α 0 [T + S + R]x, y sup[t + Sx, z Rx, t] z+ty sup[[ sup Tx, h Sx, u] Rx, t] sup [Tx, h Sx, u Rx, t] z+ty h+uz h+u+ty [T + S + R]x, y sup [Tx, h S + Rx, z] h+zy sup [Tx, h [sup Sx, u Rx, t]] sup [Tx, h Sx, u Rx, t] h+zy u+tz h+u+ty [αt + S]x, y T + S x, y [ sup T α y 1α + y 2 α y α x, y 1 α S x, y 2 α sup [αtx, y 1 αsx, y 2 ] αt + αsx, y ] Case 2 α 0 0T + 0Sx, y sup [0Tx, y 1 0Sx, y 2 ] For y 0, we have y 1 0 or y 2 0 Thus 0Tx, y 1 0Sx, y 2 0 Hence 0T + 0Sx, y 0 For y 0, we have sup [0Tx, y 1 0Sx, y 2 ] 0Tx, 0 0Sx, 0 DTx DSx DT DSx DT + Sx 0T + 0Sx, y DT + Sx if y 0 [0T + S]x, y 5 Case 1 α 0, β 0 Case 2 α 0 [αβt]x, y βt x, y y T x, [αβt]x, y α αβ [αβt]x, y DβTx if y 0 DTx if y 0 0Tx, y [αβt]x, y Case 3 α 0, β 0 [αβt]x, y βt x, y DTx if y 0 α 0Tx, y [αβt]x, y 6 It is obvious

7 S Nădăban / Filomat 30:1 2016, Fuzzy Linear Relations Definition 51 Let X, Y be two vector spaces over K A fuzzy linear relation or fuzzy linear multivalued operator between X and Y is a fuzzy linear subspace in X Y We denote by FLRX, Y the family of all fuzzy linear relations between X and Y If X Y, then we set FLRX FLRX, X If T FLRX, Y is a single-valued fuzzy function, then T will be called fuzzy linear operator We denote by FLOX, Y the family of all fuzzy linear operators between X and Y In the case X Y the family FLOX, X will be denoted FLOX Remark 52 Using Proposition 23, the linearity of T FLRX, Y can be written as Tαx 1, y 1 + βx 2, y 2 Tx 1, y 1 Tx 2, y 2, α, β K, x 1, y 1, x 2, y 2 X Y We also note that, for β 0, we obtain Tαx 1, y 1 Tx 1, y 1 Remark 53 Proposition 24 implies that Tx, y T0, 0, x X, y Y Theorem 54 Let T FRX, Y Then T FLRX, Y if and only if 1 T x1 + T x2 T x1 +x 2, x 1, x 2 X; 2 αt x T αx, x X, α K Proof 1 Let y Y Then Hence T x1 +x 2 T x1 + T x2 2 First we note that Case 1 α 0 T x1 +x 2 y Tx 1 + x 2, y Tx 1 + x 2, y 1 + y y 1 Tx 1, y 1 Tx 2, y y 1 T x1 +x 2 y sup[tx 1, y 1 Tx 2, y y 1 ] T x1 + T x2 y y 1 Y T x α if α 0 αt x y 0 if α 0, y 0 T x z : z Y} if α 0, y 0 T αx y Tαx, y T α x, y T x, y T x αt x y α α α Case 2 α 0 If y 0, as αt x y 0, we have that αt x y T αx y For y 0 we have Hence αt x y T αx y, y Y Thus αt x T αx Let x 1 + x 2, y X Y Then αt x y T x z : z Y} T0, 0 Tαx, y T αx y Tx 1 + x 2, y T x1 +x 2 y T x1 + T x2 y Tx 1 + x 2, y 1 + y 2 T x1 y 1 T x2 y 2 sup [T x1 y 1 T x2 y 2 ] On the other hand, for α 0, we have αt x αy T x y Tx, y As αt x T αx, we obtain that αt x αy T αx αy, namely Tx, y Tαx, y If α 0, we have that αt x αy T x z : z Y} Tx, z : z Y}

8 and S Nădăban / Filomat 30:1 2016, T αx αy Tαx, αy T0, 0 αt x T αx implies that Tx, z : z Y} T0, 0, namely Tx, z T0, 0, for all x X, z Y Now, we will prove that Case 1 α 0, β 0 Case 2 α 0, β 0 Tαx 1, y 1 + βx 2, y 2 Tx 1, y 1 Tx 2, y 2, α, β K, x 1, y 1, x 2, y 2 X Y Tαx 1, y 1 + βx 2, y 2 Tαx 1, αy 1 Tβx 2, βy 2 Tx 1, y 1 Tx 2, y 2 Tαx 1, y 1 + βx 2, y 2 Tβx 2, y 2 Tx 2, y 2 Tx 1, y 1 Tx 2, y 2 Case 3 α 0, β 0 Similarly to the previous case Case 4 α 0, β 0 Tαx 1, y 1 + βx 2, y 2 T0, 0 Tx 1, y 1 Tx 2, y 2 Corollary 55 If T FLRX, Y, then T 0 is a fuzzy linear subspace of Y Proof By previous theorem we have that T 0 + T 0 T 0, αt 0 T 0 This means that T 0 is a fuzzy linear subspace of Y Theorem 56 Let T FRX, Y Then T FLRX, Y if and only if T 1 FLRY, X Proof Let α, β K, y 1, x 1, y 2, x 2 Y X Then T 1 αy 1, x 1 + βy 2, x 2 T 1 αy 1 + βy 2, αx 1 + βx 2 Tαx 1 + βx 2, αy 1 + βy 2 Tαx 1, y 1 + βx 2, y 2 Tx 1, y 1 Tx 2, y 2 T 1 y 1, x 1 T 1 y 2, x 2 Similarly, we can prove that if T 1 FLRY, X, then T FLRX, Y Corollary 57 If T FLRX, Y, then T 1 0 is a fuzzy linear subspace of X Definition 58 If T FLRX, Y, then T 1 0 is called the kernel of T and it is denoted KerT Proposition 59 Let T FLOX, Y nonempty Then 1 Tx 1 + Tx 2 Tx 1 + x 2, x 1, x 2 DT; 2 αtx Tαx, x DT, α K Proof 1 T x1 + T x2 Tx 1 + Tx 2 supt x1 y 1 T x2 y 2 : y 1 + y 2 Tx 1 + Tx 2 } T x1 Tx 1 T x2 Tx 2 > 0 T x1 +x 2 Tx 1 + Tx 2 > 0, namely Tx 1 + Tx 2 Tx 1 + x 2 2 For α 0, we have αt x αtx T x Tx > 0 Thus T αx αtx > 0 Hence αtx Tαx If α 0, we must prove that T0 0, namely T0, 0 > 0 But T0, 0 Tx, y, x X, y Y, if T0, 0 0, we obtain that Tx, y, x X, y Y, ie T is the empty set Theorem 510 If T FLRX, Y, µ, µ 1, µ 2 F X and K, then 1 Tµ 1 + Tµ 2 Tµ 1 + µ 2 ;

9 2 µ 1 µ 2 Tµ 1 Tµ 2 ; 3 Tµ Tµ and Tµ Tµ, for 0 Proof 1 Let y Y fixed We will prove that S Nădăban / Filomat 30:1 2016, [Tµ 1 + Tµ 2 ]y Tµ 1 + µ 2 y If [Tµ 1 + Tµ 2 ]y 0, then the previous inequality is obvious We suppose that As A [Tµ 1 + Tµ 2 ]y A [Tµ 1 + Tµ 2 ]y > 0 for ε 0, A arbitrary, y 1, y 2 Y : y 1 + y 2 y, such that sup [Tµ 1 y 1 + Tµ 2 y 2 ], Tµ 1 y 1 + Tµ 2 y 2 > A ε But Tµ 1 y 1 sup[tx, y 1 µ 1 x] > A ε implies that there exists x 1 X such that Tx 1, y 1 > A ε, µ 1 x 1 > A ε On the other hand Tµ 2 y 2 sup[tx, y 2 µ 2 x] > A ε implies that there exists x 2 X such that Tx 2, y 2 > A ε, µ 2 x 2 > A ε Then Tx 1 + x 2, y 1 + y 2 Tx 1, y 1 Tx 2, y 2 > A ε and Thus µ 1 + µ 2 x 1 + x 2 sup x 1 +x 2 x 1+x 2 [µ 1 x 1 µ 2x 2 ] µ 1x 1 µ 2 x 2 > A ε B Tµ 1 + µ 2 y sup[tx, y µ 1 + µ 2 x] Tx 1 + x 2, y µ 1 + µ 2 x 1 + x 2 > A ε Hence B > A ε As ε is arbitrary, we obtain that B A, ie the desired inequality 2 Let y Y Then Tµ 1 y sup[tx, y µ 1 x] sup[tx, y µ 2 x] Tµ 2 y 3 Case 1 0 First, we note that On the other hand Thus x T, y 1 T x, y Tx, y x Tx, y T, y x T, y x T, y Tx, y [Tµ]y Tµ [ sup T x, y ] µx [ sup T x, y ] µx

10 S Nădăban / Filomat 30:1 2016, [ x sup T, y ] µx sup[tx, y µx] [Tµ]y Case 2 0 If y 0, then [Tµ]y 0 Thus [Tµ]y [Tµ]y If y 0, we have [Tµ]y sup z Y Tµz sup z Y As Tx, z T0, 0, x X, z Y, we obtain that sup[tx, z µx] [Tµ]y T0, 0 sup µx But [Tµ]y [Tµ]0 sup[tx, 0 µx] As 0, for x 0, we have that µx 0 Hence [Tµ]y [Tµ]y, y Y [Tµ]y T0, 0 µ0 T0, 0 sup µx Theorem 511 If T FLRX, Y, S FLRY, Z, then S T FLRX, Z Proof Let x 1, x 2 X Then Let x X and K Then S T x1 + S T x2 ST x1 + ST x2 ST x1 + T x2 ST x1 +x 2 S T x1 +x 2 S T x ST x ST x ST x S T x Proposition 512 Let T FLRX, Y nonempty Then T is single valued fuzzy function if and only if supp T 0 0} First, we note that 0 supp T 0, ie T0, 0 > 0, contrary T is empty set If there exists x 0, x supp T 0, we obtain that T 0 x > 0, namely T0, x > 0, contradiction with the fact that T is single-valued Let x DT fixed Then there exists y Y : Tx, y > 0 We suppose that T is not a single-valued fuzzy function Then there exists y Y, y y : Tx, y > 0 As we have that Tx, y Tx, y 0, contradiction Proposition 513 Let T FLRX, Y Then 1 TT 1 0 T 0; 2 T 1 T 0 T 1 0 Proof 1 First we note that Tx, y Tx, y Tx, y x, y T0, y y 0, TT 1 0 TT 1 0 y sup[tx, y T 1 x] sup[tx, y Tx, 0] 0 y sup[tx, y Tx, 0] T0, y T0, 0 T0, y On the other hand, as Tx, y Tx, 0 Tx, y x, 0 T0, y, we have 2 The proof is similar TT 1 0 y sup[tx, y Tx, 0] sup T0, y T0, y TT 1 0 y T0, y T 0y, y Y

11 S Nădăban / Filomat 30:1 2016, Proposition 514 Let T FLRX, Y nonempty Then T is injective if and only if supp KerT 0} Proof As Tx, y T0, 0, x X, y Y, we have that T0, 0 > 0, contrary T is empty set This implies that 0 supp KerT We suppose that there exists x 0 : x supp KerT, ie Tx, 0 > 0 As T is injective and x 0, we have that T x T 0, namely Tx, y T0, y 0, y Y Particulary, for y 0, we have Tx, 0 T0, 0 0, contradiction Let x 1 x 2 Then x 1 x 2 0 x 1 x 2 supp KerT, namely Tx 1 x 2, 0 0 We will prove that T x1 T x2 Indeed, for y Y, we have T x1 y T x2 y Tx 1, y Tx 2, y Tx 1, y x 2, y Tx 1 x 2, 0 0 Thus T x1 y T x2 y 0, y Y, ie T x1 T x2 T is injective Proposition 515 If T FLRX, Y, then T x T x + T 0, x X Proof Let y Y Then T x + T 0 y Hence sup [T x y 1 T 0 y 2 ] T x y T 0 0 Tx, y On the other hand T x y 1 T 0 y 2 Tx, y 1 T0, y 2 Tx, y 1 + 0, y 2 Tx, y T x + T 0 y sup [T x y 1 T 0 y 2 ] Tx, y T x + T 0 y Tx, y T x y, ie T x + T 0 T 0 Theorem 516 FLRX, Y is closed under addition Proof Let T, S FLRX, Y We will prove that T + S FLRX, Y First, we note that Hence T + S x T x + S x Using Proposition 21, we obtain T + S x y sup [T x y 1 S x y 2 ] T x + S x y T + S x1 + T + S x2 T x1 + S x1 + T x2 + S x2 T x1 +x 2 + S x1 +x 2 T + S x1 +x 2 αt + S x αt x + S x αt x + αs x T αx + S αx T + S αx Theorem 517 FLRX, Y is closed under scalar multiplication Proof Let T FLRX, Y and K We will prove that T FLRX, Y Case 1 0 As Tx, y T x, y, we obtain that Tx y T x Thus [T x1 + T x2 ]y [ 1 sup T x1 y 1 + y 2 y sup [T x1 y 1 T x2 y 2 ] [ sup T x1 1 2 ] T x2 2 ] T x2 T x1 + T x2 T x1 +x 2 T x1 +x 2 y Hence T x1 + T x2 T x1 +x 2 Now we prove that αt x T αx Case 1a α 0 Let y Y Then [αt x ]y T x T x, y y T x, α α α

12 S Nădăban / Filomat 30:1 2016, T x [αt x ] T αx T αx y α Case 1b α 0 For y 0, we have that [0T x ]y 0 Thus [0T x ]y T αx y If y 0, we have [0T x ]0 supt x z suptx, z z Y z Y DTx DTx sup Tx, z T0, 0 T0, 0 T 0x 0 z Y Hence [0T x ]y T 0x y, y Y Thus 0T x T 0x Case 2 0 Then DTx if y 0 0Tx, y [0T x1 + 0T x2 ]y On the other hand sup [0T x1 y 1 0T x2 y 2 ] 0Tx1, 0 0Tx 2, 0 if y 0 0T x1 +x 2 y 0Tx 1 + x 2, y sup [0Tx 1, y 1 0Tx 2, y 2 ] DTx1 DTx 2 if y 0 DTx1 + x 2 if y 0 We remark that DTx 1 DTx 2 DTx 1 + x 2 and thus 0T x1 + 0T x2 0T x1 +x 2 Indeed, It rest to show that α0t x 0T αx Case 2a α 0 Then On the other hand DTx 1 DTx 2 sup y 1 Y Tx 1, y 1 sup Tx 2, y 2 y 2 Y sup Tx 1, y 1 Tx 2, y 2 sup Tx 1 + x 2, y 1 + y 2 DTx 1 + x 2 y 1,y 2 Y y 1,y 2 Y [α0t x ]y 0T x α 0T αx y 0Tαx, y 0T x, y DTx if y 0 α DTαx if y 0 0Tx, y In order to establish the inclusion α0t x 0T αx, we must show that DTx DTαx But DTαx sup Tαx, y sup T α x, y sup T x, y sup Tx, y DTx α α y Y Case 2b α 0 For y 0, we have [00T x ]y 0 Thus [00T x ]y 0T 0x y If y 0, then and [00T x ]0 sup0t x z sup0tx, z 0Tx, 0 DTx z Z z Z 0T 0x 0 0T0, 0 DT0 sup T0, y T0, 0 As DTx sup Tx, y T0, 0, we obtain that [00T x ]0 0T 0x 0 Thus Hence 00T x 0T 0x [00T x ]y 0T 0x y, y Y

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