CHEN ZHI-ZHI,LIN WEIAND LUO L-LIN

(1.Institute of Mathematics,Jilin University,Changchun,130012)

(2.Institute of Statistics,Jilin University of Finance and Economics,Changchun,130017)

Projections,Birkho ffOrthogonality and Angles in Normed Spaces∗

CHEN ZHI-ZHI1,2,LIN WEI1AND LUO L-LIN1

(1.Institute of Mathematics,Jilin University,Changchun,130012)

(2.Institute of Statistics,Jilin University of Finance and Economics,Changchun,130017)

projection,norm,Birkho fforthogonality,angle,Minkowski plane,duality

1 Introduction and Preliminaries

In the research of geometry in inner spaces,orthogonality plays a very important role.In general normed spaces,many new orthogonalities have been introduced,such as Birkho ff orthogonality in[1],isosceles orthogonality in[2]and so on.These de fi nitions of orthogonalities are different,and their relations were discussed in[3].In 1993,Milicic[4]introduced g-orthogonality in normed spaces via Gateaux derivatives.In[5],it is shown that the angle A(x,y)in X satis fies the basic properties.In[6]and[7],there are more about the geometry of Minkowski plane.James[8]gave a result of Birkho fforthogonality in.

Birkho fforthogonality was introduced by Birkho ff[1]in 1935,which is the first notion of orthogonality in normed spaces.

De fi nition 1.1Letxandybe two vectors in a normed spaceX.xis said to be Birkho ff orthogonal toy,denoted byx⊥By,if for anyt∈R

In 1993,Milicic[4]introduced g-orthogonality in normed spaces via Gateaux derivatives. In fact,one has the notion of g-angle related to g-orthogonality.

De fi nition 1.2The functionalg:X2→Ris de fi ned by

The g-angle between two vectorsxandy,denoted byAg(x,y),is given by

Furthermore,xis said to be g-orthogonal toy,denoted byx⊥gy,if

In an inner product space(X,h·,·i),the angle A(x,y)between two nonzero vectors x and y in X is usually given by

⋄Parallelism:A(x,y)=0 if and only if x and y are of the same direction;A(x,y)=π if and only if x and y are of opposite direction.

⋄Symmetry:A(x,y)=A(y,x)for every x,y∈X.

⋄Homogeneity:

⋄Continuity:If xn→x and yn→y(in norm),then A(xn,yn)→A(x,y).

The g-angle is identical with the usual angle in an inner space and has the following properties:

(I)Part of parallelism property:If x and y are of the same direction,then

if x and y are of opposite direction,then

(II)Part of homogeneity property:

(III)Homogeneity property:

(IV)Part of continuity property:If yn→y(in norm),then Ag(x,yn)→Ag(x,y).

However,g-orthogonality is not equivalent to Birkho fforthogonality.In this article,we use projections to give a de fi nition of the angle Aq(x,y)between two vectors x and y such that x is Birkho fforthogonal to y if and only if

Since the angle between two vectors in a normed space is also the angle between these two vectors in the subspace spanned by them,it suffices to consider Minkowski plane,i.e.,real two dimensional normed linear space.More about the geometry of Minkowski plane could be found in[6]and[7].

Denote by L(X)the set of all bounded linear operators from X to X.For T∈L(X),the operator T′∈L(X′)is said to be the Banach conjugate operator of T if for any z∈X and any z′∈X′,there must be

2 Main Results

Let X be a Minkowski plane with a basis{e1,e2}.Suppose that x=(x1,x2)Tand y= (y1,y2)Tare two linearly independent vectors in X under the basis{e1,e2}.Put

since x and y are linearly independent.De fi ne by Pxythe projection parallel to y from X to the subspace{λx;λ∈R}.Then Pxydepends only on the vectors x and y,and has the following presentation under the basis{e1,e2}:

Proposition 2.1For any two linearly independent vectorsxandyinX,

Furthermore,denote

De fi nition 2.1For anyx,y∈X,the q-angle betweenxandyis de fi ned by

In an inner product space(X,h·,·i),obviously

and consequently,the q-angle is identical with the usual angle.

Theorem 2.1The q-angle has the following properties:

(a)Part of parallelism property:Aq(x,y)=0if and only ifxandyare linearly dependent;

(b)Part of homogeneity property:Aq(ax,by)=Aq(x,y)for everyx,y∈Xanda,b∈R{0};

(c)Continuity property:Ifxn→xandyn→y(in norm),then

Proof.(a)If x and y are linearly dependent,obviously,

If x and y are linearly independent,then

(b)Given any x,y∈X and any a,b∈R,if x and y are linearly dependent,then

if x and y are linearly independent,then

(c)Suppose that xn→x and yn→y(in norm).

If x and y are linearly independent,we may assume that xnand ynare linearly independent for each n∈N.According to the de fi nition of Pxyand the continuity of norm, the q-angle is continuous on the set{(x,y)∈X2:x and y are linearly independent}and consequently,Aq(xn,yn)→Aq(x,y).

Now consider the case of x and y being linearly dependent.By the conclusions(a)and (b),we may assume that xnand ynare linearly independent for each n∈N,xn→x and yn→ x.Fix a normal basis{e1,e2}of X.Denote x=(α,β)T,xn=(αn,βn)Tand yn=(γn,ξn)T.Consequently,αn→α,βn→β,γn→α,Notice

Lemma 2.1Letxandybe two linearly independent vectors inX.Then for anyz∈X, there exista,b∈Rsuch that

Moreover,this decomposition is unique,and

Proof.It is easy to obtained by the de fi nitions of linear independence and Pxy.

Lemma 2.2Suppose thatxis Birkho fforthogonal toy.Then for anya,b∈R,

Proof.If a=0,the conclusion is obviously true.

Theorem 2.2Letx=(x1,x2)Tandy=(y1,y2)Tbe two vectors in a Minkowski planeXwith a basis{e1,e2}.Thenxis Birkho fforthogonal toyif and only if

Proof.Sufficiency.Suppose that x is Birkho fforthogonal to y.Obviously,x and y are linearly independent.For any z∈X,we can write

by Lemma 2.1.According to Lemma 2.2,

Thus,x is Birkho fforthogonal to y.

The proof is completed.Lemma 2.3Letx=(x1,x2)Tandy=(y1,y2)Tbe two linearly independent vectors in a Minkowski planeXwith a basis{e1,e2},and letx′=(y2,−y1)Tandy′=(−x2,x1)Tbe two vectors in the dual spaceX′with the basis{δe1,δe2}.Thenis the Banach conjugate operator ofPxy,i.e.,

This equivalent description of Birkho fforthogonality incoincides with a result given by James[8]from another way.

[1]Birkho ff,G.,Orthogonality in normed linear spaces,Duke Math.J.,1(1935),169–172.

[2]James,R.C.,Orthogonality in normed linear spaces,Duke Math.J.,12(1945),291–301.

[3]Ji,D.H.and Wu,S.L.,Quantitative characterization of the di ff erence between Birkho fforthogonality and Isosceles orthogonality,J.Math.Ana.Appl.,323(2006),1–7.

[4]Milicic,P.M.,Sur le g-angle dans un espace norme,Mat.Vesnik,45(1993),43–48.

[5]Diminnie,C.R.,Andalafte,E.Z.and Freese,R.W.,Generalized angles and a characterization of inner product spaces,Houston J.Math.,14(1988),475–480.

[6]Martini,H.and Swanepoel,K.J.,The geometry of Minkowski spaces— A survey,Part I,Exposition.Math.,19(2001),97–142.

[7]Martini,H.and Swanepoel,K.J.,The geometry of Minkowski spaces— A survey,Part II,Exposition.Math.,22(2004),93–144.

[8]James,R.C.,Orthogonality and linear functionals in normed linear spaces,Trans.Amer.Math. Soc.,61(1947),265–292.

Communicated by Ji You-qing

46B20

A

1674-5647(2011)04-0378-07

date:May 24,2011.