A New View on Soft Normed Spaces

A New View on Soft Normed Spaces Tunay BILGINa , Sadi BAYRAMOVb , C ¸ igdem Gunduz(ARAS)c , Murat Ibrahim YAZARb

arXiv:1403.5064v1 [math.FA] 20 Mar 2014

a

Department of Mathematics, Yuzuncu Yıl University,Van,Turkey b Department of Mathematics, Faculty of Science and Letters, Kafkas University, TR-36100 Kars, Turkey c Department of Mathematics, Faculty of Science and Letters, Kocaeli University, TR-36100 Kocaeli, Turkey e-mails: [email protected], [email protected], [email protected], [email protected] Abstract In this paper, we work on the structure of soft linear spaces over a field K and investigate some of its properties. Here, we use the concept of the soft point which was introduced in [2, 6]. We then introduce the soft norm in soft linear spaces. Finally, we examine the properties of this soft normed space and present some investigations about soft continuous operators in the space. Key Words and Phrases. Soft norm, soft linear space, soft continuous linear operators. 1. INTRODUCTION Molodtsov [8] introduced the notion of soft set to overcome uncertainties which cannot be dealth with by classical methods in many areas such as environmental science, economics, medical science, engineering and social sciences. This theory is applicable where there is no clearly defined mathematical model. Recently, many papers concerning soft sets have been published; see [1-7] The concept of soft point was defined in different approaches. Among these, the soft point given in [2, 6] is more accurate. Also in the study [6], S.Das and et all introduced the concept of soft metric and investigated some properties of soft metric spaces. Because of the difficulties to define a vector space over a soft set based upon the concept of soft point S.Das and et all introduced the concept of soft element in [10] and defined a soft vector space by using the concept of soft element. After then they studied on soft normed spaces, soft linear operators, soft inner product spaces and their basic properties [3, 4, 9]. In this paper, by using the concept of soft point we define the soft vector space in a new point of view and investigate some of its properties. We then introduce the soft norm in soft vector spaces. Finally, we examine the properties of the soft normed space and present some investigations about soft continuous linear operators in the space. 2. PRELIMINARIES Definition 1. ([8]) A pair (F, E) is called a soft set over X ,where F is a mapping given by F : E → P (X). 1

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Definition 2. ([7]) A soft set (F, E) over X is said to be a null soft set denoted by Φ, if for all e ∈ E, F (e) = φ (null set). Definition 3. ([7]) A soft set (F, E) over X is said to be an absolute soft set ˜ if for all ε ∈ E, F (e) = X. denoted by X, Definition 4. ([5]) Let R be the set of real numbers and B(R) be the collection of all non-empty bounded subsets of R and E taken as a set of parameters. Then a mapping F : E → B(R) is called a soft real set. If a soft real set is a singleton soft set, it will be called a soft real number and denoted by r˜, s˜, t˜ etc. ˜ 0, ˜ 1 are the soft real numbers where ˜0(e) = 0, ˜1(e) = 1 for all e ∈ E , respectively. Definition 5. ([5]) Let r˜, s˜ be two soft real numbers. then the following statements (i) (ii) (iii) (iv)

˜ s if r˜≤˜ ˜ r˜≥e s if ˜ se if r˜< ˜ s˜ if r˜> hold.

r˜(e) ≤ s˜(e), for all e ∈ E ; r˜(e) ≥ s˜(e), for all e ∈ E ; r˜(e) < s˜(e), for all e ∈ E ; r˜(e) > s˜(e), for all e ∈ E ;

Definition 6. ([2, 6]) A soft set (F, E) over X is said to be a soft point if there is exactly one e ∈ E, such that F (e) = {x} for some x ∈ X and F (e′ ) = ∅, ∀e′ ∈ E/ {e}. It will be denoted by x ˜e . Definition 7. ([2, 6]) Two soft point x ˜e , y˜e′ are said to be equal if e = e′ and x = y. Thus x ˜e 6= y˜e′ ⇔ x 6= y or e 6= e′ . Proposition 1. ([2]) Every soft set can be expressed as a union of all soft points belonging to it. Conversely, any set of soft points can be considered as a soft set. ˜ be the collection of all soft points of X ˜ and R(E)∗ denote the set of Let SP (X) all non-negative soft real numbers. ˜ × SP (X) ˜ → R(E)∗ is said to be a soft Definition 8. ([6]) A mapping d˜ : SP (X) ˜ ˜ metric on the soft set X if d satisfies the following conditions: ˜ xe , y˜e )≥ ˜ ˜˜ ˜ X, 0 for all x ˜e1 , y˜e2 ∈ (M1) d(˜ 2 1 ˜ ˜ ˜ ˜ X, (M2) d(˜ xe1 , y˜e2 ) = 0 if and only if x ˜e1 = y˜e2 ∈ ˜ ˜ ˜ ˜ X, ˜e1 , y˜e2 ∈ ˜e1 ) for all x ye2 , x (M3) d(˜ xe1 , y˜e2 ) = d(˜ ˜ ye2 , z˜e3 ). ˜ xe1 , y˜e2 ) + d(˜ ˜ d(˜ ˜ xe1 , z˜e3 )≤ ˜ d(˜ ˜ X, (M4) For all x ˜e1 , y˜e2 , z˜e3 ∈ ˜ ˜ The soft set X with a soft metric d is called a soft metric space and denoted by ˜ E). ˜ d, (X, 3. Soft Normed Linear Spaces In this section, by using the concept of soft point we define the soft vector space and soft norm in a new point of view and investigate the properties of the soft normed space. Let X be a vector space over a field K (K = R) and the parameter set E be the real number set R. Definition 9. Let (F, E) be a soft set over X. The soft set (F, E) is said to be a soft vector and denoted by x ˜e if there is exactly one e ∈ E, such that F (e) = {x} for some x ∈ X and F (e′ ) = ∅, ∀e′ ∈ E/ {e} .

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˜ will be denoted by SV (X). ˜ The set of all soft vectors over X ˜ is a vector space according to the following operProposition 2. The set SV (X) ations; ˜ + y)(e+e′ ) for every x ˜e , y˜e′ ∈ SV (X); (1) x˜e + y˜e′ = (x] ˜ (2) r˜.˜ xe = (f rx)(re) for every x ˜e ∈ SV (X) and for every soft real number r˜. Definition 10. Proof. If θ ∈ X is a zero vector and e = 0 ∈ R then ˜θ0 is a soft ˜ Furthermore, (−x) f (−e) is the inverse of the soft vector x˜e . zero vector in SV (X). ˜ is a vector space . It is easy to see that the set SV (X)

˜ is called soft vector space. Definition 11. The set SV (X) 1 2 ˜ is said to be ñen of soft vectors in SV (X) Definition 12. A set S = x ˜e1 , x˜e2 , ..., x linearly independent if the following condition xn = ˜θ0 ⇔ r˜1 , r˜2 , ..., rñ = 0. x2 + ... + rñ .˜ r˜1. x˜1 + r˜2 .˜ e1

e2

en

is satisfied for the soft real numbers r˜i , 1 ≤ i ≤ n. 1 2 ˜ ñen of soft Proposition 3. A set S = x ˜e1 , x˜e2 , ..., x vectors in SV (X) is linearly 1 2 n independent if the elements of the set x , x , ..., x in X are linearly independent and the condition (r1. e1 + r2. e2 + ... + rn. en ) = 0 is satisfied. Proof. For any soft real number r˜i , 1 ≤ i ≤ n xnen = ˜θ 0 x2e2 + ... + rñ .˜ r˜1. x ˜1e1 + r˜2 .˜ ⇔ (r1. x1 + r2.^ x2 + ... + rn. xn )(r1. e1 +r2. e2 +...+rn. en ) = ˜θ0 ⇔ (r1. x1 + r2.^ x2 + ... + rn. xn ) = ˜θ and (r1. e1 + r2. e2 + ... + rn. en ) = 0 ˜ ⇔ r˜1 , r˜2 , ..., rñ = 0. ˜ be a soft vector space and M ˜ ⊂ SV (X) ˜ be a subset. Definition 13. Let SV (X) ˜ ˜ ˜ If M is a soft vector space, then M is said to be a soft vector subspace of SV (X) ˜ ˜ ˜ and denoted by SV (M )⊂SV (X). k Example 1. Let us given a class of soft vectors x ˜ek k=1,n . Then the space ) ( n X ˜kek k=1,n is a soft vector subspace. r˜i .˜ xiei generated by the class x i=1

˜ ) ⊂ SV (X) ˜ is a soft vector Example 2. If M ⊂ X is a vector subspace then SV (M subspace.

By using the definition of the soft vector, we can give the natural definition of soft norm as follows. ˜ be a soft vector space. Then a mapping Definition 14. Let SV (X) ˜ → R+ (E) k.k : SV (X) ˜ if k.ksatisfies the following conditions: is said to be a soft norm on SV (X), ˜ and k˜ ˜˜ ˜ SV (X) (N1). k˜ xe k ≥ 0 for all x ˜e ∈ xe k = ˜0 ⇔ x ˜e = ˜θ0 ;

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˜ and for every soft scalar r˜; ˜ SV (X) (N2). k˜ r .˜ xe k = |˜ r | k˜ xe k for all x ˜e ∈ ˜ ˜ k˜ ˜ SV (X). (N3). k˜ xe + y˜e′ k ≤ xe k + k˜ ye′ k for all x ˜e , y˜e′ ∈ ˜ ˜ is said to be a soft The soft vector space SV (X) with a soft norm k.k on X ˜ normed linear space and is denoted by (X, k.k). ˜ ˜ SV (X), Example 3. Let X be a normed space. In this case, for every x ˜e ∈ k˜ xe k = |e| + kxk is a soft norm. ˜ and for every soft scalar r˜; ˜ SV (X) For every x ˜e , y˜e′ ∈ ˜˜ (N1). k˜ xe k = |e| + kxk ≥ 0, k˜ xe k = ˜ 0 ⇔ |e| + kxk = ˜0 ⇔ e = 0, x = ˜θ ⇔ x ˜e = ˜θ 0

] (N2). k˜ r .˜ xe k = (r.x) r| k˜ xe k . re = |re| + kr.xk = |r| (|e| + kxk) = |˜ (N3).

k˜ xe + y˜e′ k = (x] + y)(e+e′ ) = |e + e′ | + kx + yk ≤ =

|e| + |e′ | + kxk + kyk (|e| + kxk) + (|e′ | + kyk)

k˜ xe k + k˜ ye′ k . ˜ k.k) is said to be convergent Definition 15. A sequence of soft vectors xñen in (X,

n 0 0 n ˜0λ0 as n → ∞. ˜en → x to x˜e0 ,if lim x ˜en − x˜e0 = ˜0 and denoted by x =

n→∞

˜ k.k) is said to be a Cauchy Definition 16. A sequence of soft vectors xñen in (X,

˜ ˜ ˜0 , ∃m ∈ N such that ε, ∀i, j ≥ ˜jej xiei − x

˜

i 0 as i, j → ∞. m i.e., ˜ ˜jej → ˜ xei − x

Proposition 4. Every convergent sequence is a Cauchy sequence. The proof is straight forward.

˜ k.k) be a soft normed linear space. Then (X, ˜ k.k) is said Definition 17. Let (X, ˜ converges to a soft vector of X. ˜ to be complete if every Cauchy sequence in X Definition 18. Every complete soft normed linear space is called a soft Banach space. Proposition 5. Every soft normed space is a soft metric space. ˜ xe , y˜e′ ) = ˜ k.k) be a soft normed space. If we define the soft metric by d(˜ Proof. Let (X, ˜ ˜ SV (X) then it is clear to show that the soft metric axk˜ xe − y˜e′ k for every x˜e , y˜e′ ∈ ioms are satisfied. ˜ × SV (X) ˜ → R+ (E) be a soft metric . SV (X) ˜ is a Theorem 1. Let d˜ : SV (X) soft normed space if and only if the following conditions; ˜ xe , y˜e′ ) ˜ xe + z˜e′′ , y˜e′ + z˜e′′ ) = d(˜ a) d(˜ ˜ xe , y˜e′ ) ˜ r .˜ r | d(˜ b) d(˜ xe , r˜.˜ ye′ ) = |˜ satisfied.

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˜ xe , y˜e′ ) = k˜ xe − y˜e′ k , then Proof. If d(˜ ˜ xe , y˜e′ ) ˜ xe + z˜e′′ , y˜e′ + z˜e′′ ) = k˜ xe − y˜e′ k = d(˜ xe + z˜e′′ − y˜e′ − z˜e′′ k = k˜ d(˜ and

˜ xe , y˜e′ ). ˜ r .˜ r| d(˜ r | k˜ xe − y˜e′ k = |˜ rx˜e − r˜y˜e′ k = |˜ d(˜ xe , r˜.˜ ye′ ) = k˜

˜ xe , ˜θ0 ) Suppose that the conditions of the proposition are satisfied . Taking k˜ xe k = d(˜ ˜ we have ˜ SV (X) for every x ˜e ∈ ˜ xe , θ˜0 )≥ ˜ xe , θ˜0 ) = 0˜ ⇔ x ˜ and k˜ ˜0 (N1). k˜ xe k = d(˜ xe k = d(˜ ˜e = θ˜0 ; ˜ rx ˜ rx ˜ xe , θ˜0 ) = |˜ (N2). k˜ rx ˜e k = d(˜ ˜e , θ˜0 ) = d(˜ ˜e , r˜.θ˜0 ) = |˜ r | d(˜ r | k˜ xe k ; (N3). ˜ xe , −˜ ˜ xe + y˜e′ , ˜θ0 ) = d(˜ ye′ ) k˜ xe + y˜e′ k = d(˜ ˜ ≤ =

˜ xe , ˜θ0 ) + d( ˜ ˜θ0 , −˜ d(˜ ye′ ) xe k + k˜ ye′ k . k˜ xe k + −1˜ k˜ ye′ k = k˜

˜ → SV (Y˜ ) be a soft mapping. Then T is said to be Definition 19. Let T : SV (X) soft linear operator if ˜ ˜ SV (X), ˜e , y˜e′ ∈ xe ) + T (˜ ye′ ) for every x (L1). T is additive, i.e.,T (˜ xe + y˜e′ ) = T (˜ (L2). T is homogeneous, i.e., for every soft scalar r˜, T (˜ rx ˜e ) = r˜.T (˜ xe ) for every ˜ ˜ SV (X), x˜e ∈ ˜ ˜ Definition 20. The soft operator T : SV (X) → SV (Y ) is said to be softncontinuous 0 ˜ n ˜ ˜ with x at x ˜e0 ∈SV (X) if for every sequence x ˜en of soft vectors of X ˜0e0 as ˜en → x 0 n xe0 ) as n → ∞. If T is soft continuous at each soft n → ∞, we have T (˜ xen ) → T (˜ ˜ then T is said to be soft continuous operator. vector of SV (X), ˜ → SV (Y˜ ) is said to be soft bounded, Definition 21. The soft operator T : SV (X) ˜ if there exists a soft real number M such that ˜ k˜ ˜M kT (˜ xe )k ≤ xe k , ˜ ˜ SV (X). for all x ˜e ∈ ˜ → SV (Y˜ ) is soft continuous if and only Theorem 2. The soft operator T : SV (X) if it is soft bounded. ˜ → SV (Y˜ ) be soft continuous and T is not soft Proof. Assume that T : SV (X) bounded. Thus, there exists at least one sequence xñen such that

n

T (˜ ˜ n x (3.1) xnen ) ≥˜ ˜en ,

where n ˜ is a soft real number. It is clear that xñen 6= ˜θ0 . Let us construct a soft sequence as follows; xñen

. y˜enn = n ˜ xñen n ˜ It is clear

that y˜en → θ0 as n → ∞. Since T is soft continuous, then we have

n

T (˜ ˜ as n → ∞. yen ) → 0

n

n ˜1

n

x ˜ ˜ x ˜en

en n n

T (˜

= T (˜

= ˜1,

˜ yen ) = T ) > x e n

n ˜ x ñen n ˜ x ñen n ˜ x ñen

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which is a contradiction. ˜ → SV (Y˜ ) is soft bounded and the soft Conversely, n suppose that T : SV (X) sequence x ˜en is convergent to the x ˜0e0 . In this case,

n

˜ x

T (˜ ˜M ˜0e0 → ˜0 ˜en − x ˜0e0 ) ≤ xnen − x x0e0 ) = T (˜ xnen ) − T (˜ which indicates that T is soft continuous.

˜ → SV (Y˜ ) be a soft continuous operator . Definition 22. Let T : SV (X) n o ˜ : kT (˜ ˜ k˜ ˜M kT k = inf M xe )k ≤ xe k is said to be a norm of T.

˜ kT k k˜ It is obvious that kT (˜ xe )k ≤ xe k . ˜ → SV (Y˜ ) be a soft operator. Then, Theorem 3. Let T : SV (X) kT (˜ xe )k = sup kT (˜ xe )k xe k ˜ k˜ xe k≤1 x ˜e 6=˜ θ0 k˜

kT k = sup

˜ kT k k˜ Proof. Since kT (˜ xe )k ≤ xe k , we have ˜ sup kT k k˜ ˜ kT k . sup kT (˜ xe )k ≤ xe k ≤ ˜ k˜ xe k≤1

˜ k˜ xe k≤1

Thus, ˜ kT k sup kT (˜ xe )k ≤

(3.2)

˜ k˜ xe k≤1

On the other hand, from the definition of kT k , ∀˜ε > ˜0, ∃˜ xe such that ˜ (kT k − ˜ε) k˜ kT (˜ xe )k > xe k

(3.3)

let us take y˜e = where x ˜e 6= θ˜0 . In this case, we have k˜ ye k = ˜1. If we write k˜ xe k y˜e instead of x ˜e in (3.3) we have x ˜e k˜ xe k ,

˜ (kT k − ˜ε) k˜ kT (˜ ye k˜ xe k)k > ye k˜ xe kk ⇒ (3.4)

˜ (kT k − ˜ε) k˜ kT (˜ ye )k > ye k = (kT k − ˜ε) .

If we write x ˜e instead of y˜e in (3.4), then ˜ kT k − ˜ε. sup kT (˜ xe )k > ˜ k˜ xe k≤1

Since ˜ε is arbitary we have (3.5)

˜ kT k . sup kT (˜ xe )k ≥ ˜ k˜ xe k≤1

From (3.2) and (3.5), we have kT k = sup x ˜e 6=˜ θ0

kT (˜ xe )k = sup kT (˜ xe )k . k˜ xe k ˜ k˜ xe k≤1

˜ → SV (Y˜ ) be a soft operator then kT kis a soft norm. Theorem 4. Let T : SV (X)

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˜ we have T (˜ ˜˜ ˜ SV (X) (N1) kT k ≥ 0. If kT k = ˜0, then for all x˜e ∈ xe ) = ˜θ 0 so that T = θ. (N2) k˜ r .T k = sup k˜ r .T (˜ xe )k = sup |˜ r | kT (˜ xe )k = |˜ r | sup kT (˜ xe )k = r˜. kT k

Proof.

k˜ xe k=˜ 1

k˜ xe k=˜ 1

k˜ xe k=˜ 1

(N3) kT + Sk =

xe ) + S(˜ xe )k xe )k = sup kT (˜ sup k(T + S) (˜ ˜˜ 1 k˜ xe k≤

˜˜ 1 k˜ x e k≤

˜ ≤

sup kT (˜ xe )k + sup kS(˜ xe )k = kT k + kSk . ˜˜ k˜ x e k≤ 1

˜˜ k˜ xe k≤ 1

˜ → SV (Y˜ ) and S : SV (Y˜ ) → SV (Z) ˜ be two soft Theorem 5. Let T : SV (X) operators.Then a) ˜ kSk kT k ; kS ◦ T k ≤ ˜ → SV (X) ˜ is a soft operator, then b) If T : SV (X) ˜ kT kn . kT n k ≤ is satisfied. Proof.

a) kS ◦ T k = = ˜ ≤ ˜ ≤

˜ ˜1 sup k(S ◦ T ) (˜ xe )k : k˜ xe k ≤ ˜ ˜1 sup kS(T (˜ xe ))k : k˜ xe k ≤ ˜ ˜1 sup kSk . kT (˜ xe )k : k˜ xe k ≤

kSk kT k

˜ kT k2 . Then kT n k ≤ ˜ kT kn is obtained b) If we take T = S then we have T 2 ≤ 4.

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