Vol. 24, No. 1 (2018), pp. 71–78.
2-FARTHEST ORTHOGONALITY IN GENERALIZED 2-NORMED SPACES
S. M. Mousav Shams Abad
1, H. Mazaheri
2, and M. A. Dehghan
31Faculty of Mathematics, Valiasr Rafsanjan University Rafsanjan, Iran
2Faculty of Mathematics, Yazd University Yazd, Iran
3Faculty of Mathematics, Valiasr Rafsanjan University Rafsanjan, Iran
p92356002@post.vru.ac.ir, hmazaheri@yazd.ac.ir, dehghan@vru.ac.ir
Abstract.In this paper, we consider the concepts 2-farthest orthogonality in gen- eralized 2-normed spaces. We obtain a necessary and sufficient conditions for 2- orthogonality of two elements in generalized 2-normed spaces. Also we consider-2- farthest orthogonality in generalized 2-normed spaces.
Key words and Phrases: Generalized 2-normed spaces, 2-Bounded sets, 2-Farthest orthogonality,-2-Farthest orthogonality.
Abstrak. Dalam paper ini, kita meninjau konsep Ortogonalitas Terjauh-2 di Ru- ang bernorma-2 diperumum. Kita mendapatkan syarat perlu dan syarat cukup untuk Ortogonalitas-2 dari dua unsur di ruang bernorma-2 diperumum. Kita juga meninjau Ortogonalitas Terjauh-2-di ruang bernorma-2 diperumum.
Kata kunci: Ruang bernorma-2 diperumum, Himpunan terbatas-2, Ortogonalitas terjauh-2, Ortogonalitas terjauh-2-.
1. Introduction
Approximation theory, which mainly consists of theory of nearest points (best approximation) and theory of farthest points (worst approximation), is an old and rich branch of analysis. The theory is as old as Mathematics itself. The ancient Greeks approximated the area of a closed curve by the area of a polygon. Starting in 1853, Russian mathematician P.L. Chebyshev made significant contributions in the theory of best approximation. The Weierstrass approximation theorem of 1885
2000 Mathematics Subject Classification: 46A32 , 46M05, 41A17.
Received: 04-06-2017, revised: 29-07-2017, accepted: 04-09-2017.
71
by K. Weierstrass is well known. The study was followed in the first half of the 20th Century by L.N.H. Bunt (1934), T.S. Motzkin (1935) and B. Jessen (1940).
B. Jessen was the first to make significant contributions in the theory of farthest points in 1940. This theory is very less developed as compared to the theory of best approximation (see [14]).
Let X be a real vector space with dim(X) ≥ 2. A real-valued function k., .k: X×X →Ris called a 2-norm onX if the following conditions hold:
(1)kx, yk= 0 if and only ifxandy are linearly dependent.
(2)kx, yk=ky, xkfor allx, y∈X.
(3)kαx, yk=|α|kx, ykfor allα∈Randx, y∈X.
(4)kx, y+zk ≤ kx;yk+kx;zkfor allx, y, z∈X. The pair (X;k., .k) is then called a 2-normed space.
The 2-norm concept was initially introduced by G¨ahler in 1960’s [4]. Since then, many researchers have developed and obtained various results, see for instance [5, 6, 7, 11]. Geometrically, a 2-norm function generalizes the concept of area function of parallelogram due to the fact that, in the standard case, it represents the area of the usual parallelogram spanned by the two associated vectors. Observe that in a 2-normed space we havekx, yk=kx+αy, yk for anyα∈R.
Definition 1.1. [12]Let X andY be real linear spaces. Denote by Da non-empty subset of X×Y such that for every x∈X,y∈Y the setsDx={y∈Y : (x, y)∈ D} and Dy ={x∈X : (x, y)∈D} are linear subspaces of the spaces Y andX, respectively. A function k., .k: D →[0,∞) will be called a generalized 2-norm on D if it satisfies the following conditions:
(1)kx, αyk=|α|.kx, yk=kαx, yk for any real numberαand all (x, y)∈D;
(2)kx, y+zk ≤ kx, yk+kx, zk forx∈X,y, z∈Y with(x, y),(x, z)∈D;
(3)kx+y, zk ≤ kx, zk+ky, zk for x, y∈X, z ∈Y with (x, z),(y, z) ∈D.
The setD is called a 2-normed set.
In particular, ifD =X×Y, the functionk., .kis said to be a generalized 2- norm onX×Y and the pair (X×Y,k., .k) is called a generalized 2-normed space. If X =Y , then the generalized 2-normed space (X×X,k., .k) is denoted by (X,k., .k).
In the case that X =Y and D =D−1, where D−1 ={(y, x) : (x, y) ∈D}, and kx, yk=ky, xkfor all (x, y)∈D, we callk., .ka generalized symmetric 2-norm and D a symmetric 2-normed set.
Recall that in G¨ahler definition of a 2-normkx, yk= 0 if and only ifxandy are linearly dependent, and this is a crucial difference between G¨ahlerfs approach and Lewandowska’s one.
Example 1.2. [12] Let X be a real linear space having two seminorms k.k1 and k.k2. Then(X,k., .k) is a generalized2-normed space with the2-norm defined by
kx, yk=kxk1.kyk2; x, y∈X.
Every 2-normed space is a locally convex topological vector space. In fact for a fixed b ∈ X, pb = kx, bk; x ∈ X is a semi-norm on X and the family
P ={pb :b∈X} of semi-norms generates a locally convex topology. We know in a topological vector space (X, τ) over a fieldF, W is called bounded if for every neighborhoodN of the zero vector there exists a scalarαsuch that
W ⊆αN withαN :={αx|x∈N}.
In 2-normed spaces , the setW is 2-bounded set ifσ(M)<∞, where σ(M) = sup{kx−z, y−zk: x, y, z∈M}.
Let (X,k., .k) be a 2-normed space and letW1 and W2 be two subspaces of X. A mapf : W1×W2→Ris called a bilinear 2-functional onW1×W2whenever for allx1, x2∈W1,y1, y2∈W2 and allλ1, λ2∈R;
(i)f(x1+x2, y1+y2) =f(x1, y1) +f(x1, y2) +f(x2, y1) +f(x2, y2), (ii)f(λ1x1, λ2y1) =λ1λ2f(x1, y1).
A bilinear 2-functionalf : W1×W2→Ris called bounded if there exists a non-negative real numberM(called a Lipschitz constant forf) such that|f(x, y)| ≤ Mkx, ykfor allx∈W1 and ally∈W2. Also, the norm of a bilinear 2-functional f is defined by
kfk= inf{M ≥0 : M is a Lipschitz constant f}.
It is known that
kfk = sup{|f(x, y)|: (x, y)∈W1×W2, kx, yk ≤1}
= sup{|f(x, y)|: (x, y)∈W1×W2, kx, yk= 1}
= sup{|f(x, y)|
kx, yk : (x, y)∈W1×W2, kx, yk>0}.
For a generalized 2-normed space (X,k., .k) and 06= b ∈ X, we denote by Xb∗, the Banach space of all bounded bilinear 2-functionals onX×< b >, where
< b >be the subspace ofX generated byb.
2. 2-Farthest Orthogonality in Generalized 2-Normed Spaces In this section we consider farthest orthogonality in generalized 2-normed spaces.
Definition 2.1. Let (X,k., .k)be a generalized 2-normed space. For x, y∈X, we say thatxis2-farthest orthogonality toy and denote byx⊥2F y , if for allz∈X,
kx, zk ≥ kx−y, zk andkx−y, zk 6= 0.
IfW a subset ofX andx∈X, we say thatx⊥2FW if and only ifx⊥2Fy.for every y∈W.
Let (X,k., .k) be a generalized 2-normed space and W be bounded subset of X. A pointy0∈W is said to be a 2-farthest point forx∈X,if
kx−y0, zk ≥ kx−y, zkf or all z∈X and f or all y∈W.
If fW(x, z) = supy∈Wkx−y, zk, then y0 is 2-farthest point of x, if kx−y0, zk = fW(x, z) for allz∈X. For ally∈W,put
FW2 (x) ={y0∈W : kx−y0, zk=fW(x, z)f or all z∈X}.
Forx∈X, ifFW2(x) is nonempty, we say thatW is 2-remotal. Forx∈X, if FW2(x) is singleton, we say thatW is 2-uniquely remotal. Forx∈X, ifFW2 (x) =∅, we say thatW is 2-anti-remotal.
A sequence {xn}n≥1 in a generalized 2-normed space (X,k., .k) is called a convergent sequence if there exists anx∈X such thatlimn→∞kxn, zk=kx, zkfor allz∈X. If{xn}n≥1 converges tox, we writexn→xasn→ ∞.
Theorem 2.2. Let(X,k., .k)be a generalized2-normed space andx, y∈X. If for somek∈R\{0},y=kxandx⊥2F y, then0≤k≤2.
Proof. Supposey=kx, Thereforekx, zk ≥ k(1−k)x, zkf or all z∈X. It follows that 1≥ |1−k|. Hence 0≤k≤2.
Theorem 2.3. Let(X,k., .k)be a generalized 2-norm andx, t∈X. For allz∈X, ifkx, zk ≥ kx−t, zk, then for everyk≥1,kkx, zk ≥ kkx−t, zk, that is, ifx⊥2Ft, then for every k≥1,kx⊥2Fy.
Proof. Supposez∈X, defineFz: [0,∞)→RbyFz(k) =kkx−t, zk − kkx, zk.
Then Fz is convex function such that Fz(0) = kt, zk ≥ Fz(k) for every k ∈ R, Fz(1) =kx−t, zk − kx, zk ≤0. Thus we haveFz(k)≤0 for everyk≥1.
Example 2.4. SupposeX =R2with the2-normk., .kdefined byk(a1, a2),(b1, b2)k=
|a1b2−a1b1| is a 2-norm. Then(3,3)⊥2F(2,2).
Proposition 2.5. Let (X,k., .k)be a 2-normed space. Then:
(i) Ifx∈X,then x⊥2F0.
(ii) Ifx⊥2Fy andkx, zk=ky, zk, for allz∈X, theny⊥2Fx.
(iii) If0⊥2Fxforx∈X, thenx= 0.
(vi) Forα∈R,x⊥2Fy if and only if αx⊥2Fαy.
(v) Ifxn→x,yn →y andxn⊥2Fyn, thenx⊥2Fy.
Proof. It is trivial.
Theorem 2.6. Let (X,k., .k) be a generalized 2-norm and x, y ∈ X. Then the following statements are equivalent:
(1)x⊥2Fy,
(2)for every z ∈ X, there exists a 2-bilinear functional Tz ∈ Xz∗ such that
|Tz(x, z)| ≥ kx−y, zk andkTzk= 1.
Proof. (2)⇒(1). For everyz∈X, suppose that there is a 2-bilinear functional Tz∈Xz∗such that |Tz(x, z)| ≥ kx−y, zk andkTzk= 1. Then
kx, zk = kx, zkkTzk
≥ |Tz(x, z)|
≥ kx−y, zk.
(1) ⇒ (2). Conversely suppose thatx⊥2Fy and z ∈ X. For all z ∈ X, we have kx−y, zk 6= 0, it follows that kx, zk 6= 0. From Hahan Banach Theorem in the context of 2-normed spaces (see Theorem 2.2 [12]), there exists a bilinearTz∈Xz∗, such thatkTzk= 1 and|Tz(x, z)|=kx, zk ≥ kx−y, zk.
Theorem 2.7. Let (X,k., .k)be a generalized 2-norm, W a bounded subset of X andx∈X. If for all z∈X, there exists a2-bilinear functionalTz∈Xz∗ such that
|Tz(x, z)|=fW(x, z)andkTzk= 1, thenx⊥2FW,
Proof. Forz∈X andy∈W,fW(x, z)≥ kx−y, zk. From Theorem 2.2,x⊥2Fy.
Thereforex⊥2FW.
Theorem 2.8. Let (X,k., .k) be a generalized 2-norm andW a bounded subset of X. Ifg1∈FW2(x),g2∈FW2 (y)andz∈X,
(i)2kx−y, zk ≤ kx−g1, zk+ky−g2, zk, (ii) Ifg1−W =W, then x−g1⊥2FW.
Proof. (i) From definition of 2-farthest points, we have for everyz∈X kx−y, zk ≤ kx−g1, zk, kx−y, zk ≤ ky−g1, zk.
Therefore 2kx−y, zk ≤ kx−g1, zk+ky−g2, zk.
(ii) Fory∈W,
kx−g1, zk ≥ kx−y, zk
= kx−g1+g1−y, zk.
Supposeg1−y=u, thenu∈W. Thereforex−g1⊥2FW.
Theorem 2.9. Let (X,k., .k)be a generalized 2-norm, W a bounded subset of X andg0∈W. Then the set (F2W)−1(g0) ={x∈X : g0∈FW2 (x)} is closed.
Proof. Suppose {xn}n≥1 any sequence in (F2W)−1(g0) and x ∈ X such that limn→∞xn=x. Then for everyz∈X andg∈W and forn≥1 , we have
kxn−g0, zk ≥ kxn−g, zk.
By limiting to ∞, we have for every z ∈ X and for every g ∈ W kx−g0, zk ≥ kx−g, zk. That isx∈(F2W)−1(g0), and (F2W)−1(g0) is closed.
Corollary 2.10. Let (X,k., .k) be a generalized 2-norm and y0 ∈X. Put Ey0 = {x∈X : x⊥2Fy0}, then Ey0 is closed inX.
Proof. Suppose{xn}n≥1any sequence inEy0 andx∈X such thatlimn→∞xn= x. Therefore for everyz ∈X,limn→∞kxn, zk=kx, zk. Sincexn⊥2Fy0, for every n ≥ 1, kxn, z| ≥ kxn −y0, zk. Therefore kx, zk ≥ kx−y0, zk. It follows that x⊥2Fy0.
3. -2-Farthest Orthogonality
In this section we consider-2-farthest points in generalized 2-norm spaces.
Definition 3.1. Let (X,k., .k) be a generalized 2-norm space and x, y ∈ X and >0. We say thatx⊥2Fy, if for all z∈X,
kx, zk ≥ kx−y, zk −, andkx−y, zk 6=.
Also, if W is a subset of X and x∈ X, then x⊥2FW, if x⊥2Fy for every y∈W. Put
FW,2 (x) ={y0∈W : kx−y0, zk ≥fW(x, z)− f or all z∈X}.
Theorem 3.2. Let (X,k., .k)be a generalized 2-norm, x, y∈X and >0. Then the following statements are equivalent:
(1)x⊥2Fy,
(2) for all z ∈ X, there exists a 2-bilinear functional Tz ∈ Xz∗ such that
|Tz(x, z)| ≥ kx−y, zk −andkTzk= 1.
Proof. (2)⇒(1). For everyz∈X, suppose that there is a 2-bilinear functional Tz∈Xz∗such that |Tz(x, z)| ≥ kx−y, zk −andkTzk= 1. Then
kx, zk = kx, zkkTzk
≥ |Tz(x, z)|
≥ kx−y, zk −.
(1) ⇒ (2). Conversely, suppose x⊥2Fy and z ∈ X. For all z ∈ X, we have kx−y, zk 6= , it follows that kx, zk 6= 0. From Hahan Banach Theorem in the context of 2-normed spaces (see Theorem 2.2 [12]), there exists a bilinearTz∈Xz∗, such thatkTzk= 1 and|Tz(x, z)|=kx, zk ≥ kx−y, zk −.
Theorem 3.3. Let (X,k., .k) be a generalized 2-norm, W a subset of X, x∈ X and >0. If for all z∈X, there exists a 2-bilinear functional Tz∈Xz∗ such that
|Tz(x, z)|=fW(x, z)−andkTzk= 1, then x⊥2FW,
Proof. Forz∈X andy∈W,fW(x, z)−≥ kx−y, zk −. From above Theorem, x⊥2Fy. Thereforex⊥2FW.
Theorem 3.4. Let (X,k., .k) be a generalized 2-norm and W a subset of X. If g1∈FW,2 (x),g2∈FW,2 (y)andz∈X, then
(i)2(kx−y, zk −)≤ kx−g1, zk+ky−g2, zk, (ii) Ifg1−W =W, then x−g1⊥2FW.
(iii) Forα≥0,x⊥2Fy if and only ifαx⊥α2Fαy Proof. It is trivial.
Definition 3.5. Let (X,k., .k) be a generalized 2-norm and G and A be a non- empty subset ofX,z∈X. A pointg0∈Gis called a simultaneous z-farthest point toA fromGif
ρz(A, G) = sup
g∈G
sup
a∈A
kg−a, zk= sup
a∈A
kg0−a, zk.
Put
FGz(A) ={g∈G: sup
a∈A
kg−a, zk=ρz(A, G)}.
Theorem 3.6. Let(X,k., .k)be a generalized 2-norm andGandAbe a non-empty subset ofX,z∈X. Ifg∈FGz(A)∩L(y, W)for somew, y∈G. Theny, w∈FGz(A), whereL(y, W) ={ty+ (1−t)W : 0< t <1}.
Proof. Supposeg∈FGz(A)∩L(y, w) for somew, y∈G. Then supa∈Akg−a, bk= ρz(A, G) andg=ty+ (1−t)wfor somet∈[0,1]. Further
sup
a∈A
kg−a, bk = kty+ (1−t)w−a, zk
= sup
a∈A
kt(y−a) + (1−a)(w−a), zk
≤ tsup
a∈A
ky−a, zk+ (1−t) sup
a∈A
kw−a, zk
≤ tρz(A, G) + (1−t)ρz(A, G)
= ρz(A, G).
We assume that supa∈Aky−a, zk < ρz(A, G) or supa∈Akw−a, zk < ρz(A, G).
Thenρz(A, G)< ρz(A, G), which is contradicition. Therefore sup
a∈A
ky−a, zk=ρz(A, G) = sup
a∈A
kw−a, zk= sup
a∈A
kg−a, zk.
Thereforey, w∈FGz(A).
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