SOME RESULTS ON 2-INNER PRODUCT SPACES

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Vol. 37, No. 2, 2007, 35-40

SOME RESULTS ON 2-INNER PRODUCT SPACES

H. Mazaheri¹, R. Kazemi¹

Abstract. We onsider ”Riesz Theorem” in the 2-inner product spaces and give some results in this field. Also, we give some characterizations about 2-inner product spaces in b-approximation theory.

AMS Mathematics Subject Classification (2000): 41A65, 41A15

Key words and phrases: b-Orthogonality, 2-Normed spaces, 2-Inner product, b-Proximinal subspaces, b-Best approximation

1. Introduction

The concept of linear 2-normed spaces has been investigated by S. G¨ahler (see [9]) and has been developed extensively in different subjects by many au- thors (see [1-8]).

Let X be a linear space of dimension greater than 1. Suppose k., .k is a real-valued function on X×X satisfying the following conditions:

a)kx, yk= 0 if and only if xand yare linearly dependent vectors.

b) kx, yk=ky, xkfor allx, y∈X.

c) kλx, yk=|λ|kx, ykfor allλ∈Rand allx, y ∈X.

d) kx+y, zk ≤ kx, zk+ky, zkfor allx, y, z∈X.

Then k., .k is called a 2-norm on X and (X,k., .k) is called a linear 2-normed space. Some of the basic properties of 2-norms are that they are non-negative andkx, y+αxk=kx, yk for allx, y∈X and allα∈R.

Every 2-normed space is a locally convex topological vector space. In fact, for a fixed b ∈ X, pb(x) = kx, bk, x∈ X, is a seminorm and the family P = {pb: b∈X}of seminorms generates a locally convex topology on X.

Let (X,k., .k) be a 2-normed space and let W1 and W2 be two linear subspaces of X. A map Λ : W1×W2 → R is called a bilinear 2-functional on W₁×W₂, whenever for all x₁, x₂∈W₁, y₁, y₂∈W₂andλ₁, λ₂∈R;

a)Λ(x1+x2, y1+y2) = Λ(x1, y1) + Λ(x1, y2) + Λ(x2, y1) + Λ(x2, y2), b) Λ(λ1x1, λ2y1) =λ1λ2Λ(x1, y1).

A bilinear 2-functional Λ : W1×W2 → R is said to be bounded if there exists a non-negative real number M (called a Lipschitz constant for Λ) such that |Λ(x, y)| ≤ Mkx, yk for all x∈W1 and ally ∈W2. Also, the norm of a bilinear 2-functional Λ is defined by

kΛk= inf{M ≥0 : M is a Lipschitz constant f or Λ}.

1Department of Mathematics, Yazd University, Yazd, Iran, e-mail : [email protected]

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It is known that ([4])

kΛk = sup{|Λ(x, y)|: (x, y)∈W1×W2, kx, yk ≤1}

= sup{|Λ(x, y)|: (x, y)∈W1×W2, kx, yk= 1}

= sup{|Λ(x, y)|/kx, yk: (x, y)∈W1×W2, kx, yk>0}.

For a 2-normed space (X,k., .k) and 06=b∈X, byX_b^∗is denoted the Banach space of all bounded bilinear 2-functionals onX×, where is the subspace ofX generated byb.

Let (X,k., .k) be a 2-normed space and x, y ∈ X, then x is said to be orthogonal to y if and only if there exists b ∈ X such that for all scalar α, kx, bk 6= 0 andkx, bk ≤ kx+αy, bk, in this case we writex⊥^by. If M1 andM2

are subsets ofX, we say thatM1is orthogonal toM2if and only if there exists b ∈X such thatg1⊥^bg2 for all g1 ∈M1, g2∈M2. If M1 is orthogonal to M2, we writeM1⊥^bM2. (see [10])

Let (X,k., .k) be a 2-normed space, x ∈ X, A be a linear subspace of X and b ∈ X\x−A. y0 ∈ A is b-best approximation for x∈ X, if x−y0⊥^bA.

Therefore,y0∈Ais a b-best approximation ofxif for all y∈A kx−y0, bk ≤ kx−y, bk,

thenkx−y0, bk= infy∈Akx−y, bk=kx−A, bk. The set of all b-best approx- imations of xin A is denoted by P_A^b(x). A is called b-proximinal if for every x∈X\(A+) there exist y0∈A such thaty0 ∈P_A^b(x). Also,A is called b-Chebyshev if for everyx∈X\(A+), there exists a uniquey0∈Asuch thaty0∈P_A^b(x).

LetX be a linear space of dimension greater than 1 over the fieldK=Rof real numbers or the field K=Cof complex numbers. Suppose that (., .|.) is a K-valued function defined onX×X×X satisfying the following conditions:

a)(x, x|z)≥0 and (x, x|z) = 0 if and only ifxandzare linearly dependent;

b)(x, x|z) = (z, z|x);

c)(y, x|z) = (x, y|z);

d)(αx, y|z) =α(x, y|z) for any scalar α∈K;

e)(x+ ´x, y|z) = (x, y|z) + (´x, y|z).

(., .|.) is called a 2-inner product onX and (X,(., .|.)) is called a 2-inner product space. Some basic properties of 2-inner products (., .|.) can be immediately obtained [1-3].

Let (X,(., .|.)) be a 2-inner product space. We can define a 2-norm onX×X by

kx, yk=p

(x, x|y).

Let (X,(., .|.)) be a 2-inner product space,b∈X and x, y∈X\. Then x⊥^by⇔(x, y|b) = 0.

Using the above properties, we can prove the Cauchy-Schwartz inequality (x, y|b)²≤ kx, bk²ky, bk²

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for everyx, y∈X. Moreover, the equality holds in this inequality if and only if xandy are linearly dependent. Also, we have the parallelogram-law

kx+y, bk²+kx−y, bk²= 2kx, bk²+ 2ky, bk²

for every x, y∈X (For more details about 2-inner product space see [1-3]).

2. Main results

In this section we shall obtain some characterization of 2-inner product spaces.

Theorem 2.1. Let (X,(., .|.))be a 2-inner product space,b∈X andΛ∈X_b^∗. If the set M ={x∈ X : (x, b) ∈ kerΛ} is b-proximinal, then there exists a y∈X such that

Λ(x, b) = (x, y|b) ∀x∈X.

Proof. If Λ = 0, put y= 0.

If Λ6= 0, there existsx1∈X such that Λ(x1, b)6= 0. SinceM is a b-proximinal, there existsm∈M such thatx2=x1−m⊥^bM andkx1−m, bk 6= 0. Therefore, (x2, y|b) = 0 for ally∈M. Put z= _kx^x²

2,bk. Then, (z, y|b) = 0 andkz, bk= 1.

For allx∈X, we setu= Λ(x, b)z−Λ(z, b)x. Then, Λ(u, b) = Λ(x, b)Λ(z, b)−

Λ(z, b)Λ(x, b) = 0. It follows thatu∈M, therefore, (z, u|b) = 0. Now 0 = (z, u|b) = (Λ(x, b)z−Λ(z, b)x, z|b)

= Λ(x, b)(z, z|b)−Λ(z, b)(x, z|b).

Hence, (z, z|b)Λ(x, b) = Λ(z, b)(x, z|b) and Λ(x, b) = (x, y|b), wherey=zΛ(z, b).

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Definition 2.2. Let (X,(., .|.))be a 2-inner product space, b∈X.

a) A sequence {xn} inX is a b-Cauchy sequence if

∀² >0 ∃N >0, such that∀ m, n ≥ N 0<kxm−xn, bk< ²

b) X is b-Hilbert if every b-Cauchy sequence is converges in the seminormed (X,k., bk).

c) If a subset A in X is closed in the space (X,k., bk), then we say that A is b-closed in the seminormed (X,k., bk).

Theorem 2.3. Let (X,(., .|.)) be a 2-inner product space, A be a convex set in X andb∈X. Then each x∈X\A+has at most one b-best approxi- mation inA.

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Proof. Suppose x ∈ X\A+  and y1, y2 ∈ P_A^b(x). By convexity A,

1

2(y1+y2)∈A. Therefore

kx−A, bk ≤ kx−1

2(y1+y2), bk

= k1

2(x−y1) +1

2(x−y2), bk

≤ 1

2kx−y1, bk+1

2kx−y2, bk

= 1

2kx−A, bk+1

2kx−A, bk

= kx−A, bk.

Hence equality must hold throughout these inequalities. By the condition of equality and b-strictly convex in the triangle inequality,x−y1=k(x−y2) for somek≥0. But,kx−y1, bk=kx−A, bk=kx−y2, bkimpliesk= 1, and hence

y1=y2. 2

Theorem 2.4. Every nonempty b-closed, convex A in a b-Hilbert space X withA∩=φ, is b-Chebyshev.

Proof. Supposex∈X\A+, putE =x−Aandδ= infy∈Eky, bk. Then E is a b-closed and convex set inX. Supposey0, y∈E, sinceE is convex and

y0+y

2 ∈E, therefore using the parallelogram-law:

1

2ky0 −y, bk²≤ ky0, bk²+ky, bk²−2δ². (∗)

Let {yn} be a sequence in E, where kyn, bk −→ δ. From (∗), since for all m, n≥ 1, kyn −ym, bk 6= 0 then {yn} is a b-Cauchy sequence. Since X is b- Hilbert, there existy0∈ X such thatyn −→y0 as n → ∞, alsoE is b-closed andyn∈E, thereforey0∈E. It follows thatky0, bk=δ. Therefore, there exists a0∈Asuch thatkx−a0, bk=kx−A, bk. That is,a0is a b-best approximation forx.

For uniqueness, if a1, b1 ∈ A are b-best approximations for x. Put y1 = x−a1, y2=x−b1, thenky1, bk=ky2, bk=δ. If we apply the inequality (∗), it

follows thaty1=y2, thereforea1=b1. 2

Let (X,(., .|.)) be a 2-inner product space,Abe a subspace ofX andb∈X.

Put

A^⊥_b ={x∈X : (x, g|b) = 0,∀g∈A}.

Theorem 2.5. Let Abe a b-Chebyshev subspace of the 2-inner product space (X,(., .|.)),b∈X (e.g. a closed subspace of a b-Hilbert space) and AT

=

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∅. ThenA^⊥_b is a b-Chebyshev subspace, and the following statements are true:

a) X=A^⊥_b ⊕A

b) A={x∈X : kx, bk=kx−A^⊥_b, bk}

c) A^⊥_b ={x∈X : kx, bk=kx−A, bk}

d) kx, bk²=kg, bk²+kg0, bk², for allx∈X\, where x=g+g0, g∈A andg0∈A^⊥

Proof. If x∈X, then there exists y ∈A such that x−y⊥^bA. Puty0=x−y thenx=y+y0 andy0∈A^⊥_b, impliesX =A+A^⊥_b and

kx−A^⊥_b, bk = ky−A^⊥_b, bk

≤ ky, bk.

Now ifz∈A^⊥_b andy∈A, we have

kx−A^⊥_b, bk² ≥ kz−y, bk²

= (z−y, z−y|b)

= kz, bk²+ky, bk² (∗∗)

≥ ky, bk². Thereforekx−y0, bk=kx−A^⊥_b, bk, i.e.,y0∈P_A^b⊥

b(x).

Ifg0, y0 ∈P_A^b⊥

b(x), thenx=y+y0 =g+g0 for somey, g ∈A. It follows that y0−g0 ∈ A^⊥_b ∩A ={0}, hencey0 = g0. Therefore A^⊥_b is b-Chebyshev, X =A⊕A^⊥_b ,ky−A^⊥_b , bk=ky, bk and from (∗∗) we have

kx, bk²=ky, bk²+ky0, bk². If x∈X and kx−A^⊥_b, bk =kx, bk, then 0∈P_A^b⊥

b(x) and x=y+y0 for some y∈A andy0∈A^⊥_b. Hence

kx−A^⊥_b, bk = ky−A^⊥_b, bk

= ky, bk

= kx−y0, bk.

impliesy0∈P_A^b⊥

b(x). Thereforey0= 0 and x=y∈A. Then A={x∈X : kx, bk=kx−A^⊥_b, bk}.

Finally, by paying attention to the definition of A^⊥_b we have A^⊥_b ={x∈X : kx, bk=kx−A, bk}.

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Received by the editors June 6, 2006