Open Access

On strong orthogonality and strictly convex normed linear spaces

Journal of Inequalities and Applications20132013:242

https://doi.org/10.1186/1029-242X-2013-242

Received: 19 November 2012

Accepted: 25 April 2013

Published: 14 May 2013

Abstract

We introduce the notion of a strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element x of the unit sphere S X to be an exposed point of the unit ball B X . We then prove that a normed linear space is strictly convex iff for each element x of the unit sphere, there exists a bounded linear operator A on X which attains its norm only at the points of the form λx with λ S K .

MSC:46B20, 47A30.

Keywords

orthogonalitystrict convexityextreme point

1 Introduction

Suppose ( X , ) is a normed linear space over the field K, real or complex. X is said to be strictly convex iff every element of the unit sphere S X = { x X : x = 1 } is an extreme point of the unit ball B X = { x X : x 1 } . There are many equivalent characterizations of the strict convexity of a normed space, some of them given in [1, 2] are as follows.
  1. (i)

    If x , y S X , then we have x + y < 2 .

     
  2. (ii)

    Every non-zero continuous linear functional attains a maximum on at most one point of the unit sphere.

     
  3. (iii)

    If x + y = x + y , x 0 , then y = c x for some c 0 .

     

The notion of strict convexity plays an important role in the studies of the geometry of Banach spaces. One may go through [111] for more information related to strictly convex spaces.

An element x is said to be orthogonal to y in X in the sense of Birkhoff-James [1, 8, 12], written as, x B y , iff
x x + λ y for all scalars  λ .

If X is an inner product space, then x B y implies x < x + λ y for all scalars λ 0 . Motivated by this fact, we here introduce the notion of strong orthogonality as follows.

Strongly orthogonal in the sense of Birkhoff-James: In a normed linear space X, an element x is said to be strongly orthogonal to another element y in the sense of Birkhoff-James, written as x S B y , iff
x < x + λ y for all scalars  λ 0 .

If x S B y , then x B y , but the converse is not true. In l ( R 2 ) the element ( 1 , 0 ) is orthogonal to ( 0 , 1 ) in the sense of Birkhoff-James, but not strongly orthogonal.

Strongly orthogonal set relative to an element: A finite set of elements S = { x 1 , x 2 , , x n } is said to be a strongly orthogonal set relative to an element x i 0 contained in S in the sense of Birkhoff-James iff
x i 0 < x i 0 + j = 1 , j i 0 n λ j x j

whenever not all λ j ’s are 0.

An infinite set of elements is said to be a strongly orthogonal set relative to an element contained in the set in the sense of Birkhoff-James iff every finite subset containing that element is strongly orthogonal relative to that element in the sense of Birkhoff-James.

Strongly orthogonal set: A finite set of elements { x 1 , x 2 , , x n } is said to be a strongly orthogonal set in the sense of Birkhoff-James iff for each i { 1 , 2 , , n }
x i < x i + j = 1 , j i n λ j x j

whenever not all λ j ’s are 0.

An infinite set of elements is said to be a strongly orthogonal set in the sense of Birkhoff-James iff every finite subset of the set is a strongly orthogonal set in the sense of Birkhoff-James.

Clearly if a set is strongly orthogonal in the sense of Birkhoff-James, then it is strongly orthogonal relative to every element of the set in the sense of Birkhoff-James. If X has a Hamel basis which is strongly orthogonal in the sense of Birkhoff-James, then we call the Hamel basis a strongly orthogonal Hamel basis in the sense of Birkhoff-James, and if X has a Hamel basis which is strongly orthogonal relative to an element of the basis in the sense of Birkhoff-James, then we call the Hamel basis a strongly orthogonal Hamel basis relative to that element of the basis in the sense of Birkhoff-James. If, in addition, the norm of each element of a strongly orthogonal set is 1, then accordingly we call them orthonormal.

As, for example, the set { ( 1 , 0 , , 0 ) , ( 0 , 1 , 0 , , 0 ) , , ( 0 , 0 , , 1 ) } is a strongly orthonormal Hamel basis in the sense of Birkhoff-James in l 1 ( R n ) , but not in l ( R n ) .

In l 2 ( R 3 ) the set { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 1 , 1 ) } is strongly orthogonal relative to ( 1 , 0 , 0 ) in the sense of Birkhoff-James, but not relative to ( 0 , 1 , 1 ) .

In this paper we give another characterization of strictly convex normed linear spaces by using the Hahn-Banach theorem and the notion of a strongly orthogonal Hamel basis relative to an element in the sense of Birkhoff-James. More precisely, we explore the relation between the existence of a strongly orthogonal Hamel basis relative to an element with the unit norm in the sense of Birkhoff-James in a normed space and that of an extreme point of the unit ball in the space. We also prove that a normed linear space is strictly convex iff for each point x of the unit sphere, there exists a bounded linear operator A on X which attains its norm only at the points of the form λx with λ S K .

2 Main results

We first obtain a sufficient condition for an element in the unit sphere to be an extreme point of the unit ball in an arbitrary normed linear space.

Theorem 2.1 Let X be a normed linear space and x 0 S X . If there exists a Hamel basis of X containing x 0 which is strongly orthonormal relative to x 0 in the sense of Birkhoff-James, then x 0 is an extreme point of B X .

Proof Let D = { x 0 , x α : α Λ } be a strongly orthonormal Hamel basis relative to x 0 in the sense of Birkhoff-James.

If possible, suppose that x 0 is not an extreme point of B X , then x 0 = t z 1 + ( 1 t ) z 2 where 0 < t < 1 and z 1 = z 2 = 1 .

So, there exists α 1 , α 2 , , α n in Λ such that
z 1 = β 0 x 0 + j = 1 n β j x α j and z 2 = γ 0 x 0 + j = 1 n γ j x α j

for some scalars β j , γ j ( j = 0 , 1 , 2 , , n ).

If β 0 = 0 and γ 0 = 0 , then x 0 = t z 1 + ( 1 t ) z 2 implies that
x 0 = j = 1 n ( t β j + ( 1 t ) γ j ) x α j ,

which contradicts the fact that every finite subset of D is linearly independent. So, the case β 0 = 0 and γ 0 = 0 is ruled out.

If β 0 0 , γ 0 = 0 , then as { x 0 , x α 1 , x α 2 , , x α n } is a strongly orthonormal set relative to x 0 in the sense of Birkhoff-James, so we get
1 = z 1 = | β 0 | x 0 + j = 1 n β j β 0 x α j | β 0 | .
Now
x 0 = t β 0 x 0 + j = 1 n ( t β j + ( 1 t ) γ j ) x α j

and so t β 0 = 1 , which is not possible as | β 0 | 1 and 0 < t < 1 .

Similarly β 0 = 0 , γ 0 0 is also ruled out.

Thus we have β 0 0 and γ 0 0 .

Our claim is that at least one of | β 0 | , | γ 0 | must be less than 1.

If possible, suppose that | β 0 | > 1 . Then
β 0 x 0 + j = 1 n β j x α j = | β 0 | x 0 + j = 1 n β j β 0 x α j | β 0 | > 1 .

This contradicts z 1 = 1 . Thus | β 0 | 1 . Similarly | γ 0 | 1 . We next show that | β 0 | = 1 and | γ 0 | = 1 cannot hold simultaneously.

Case 1. X is a real normed linear space.

Then | β 0 | = 1 implies that
1 = z 1 = | β 0 | x 0 + j = 1 n β j β 0 x α j > x 0 ,

unless β i = 0 i = 1 , 2 , , n .

Thus | β 0 | = 1 z 1 = β 0 x 0 z 1 = ± x 0 x 0 = z 1 = z 2 or t = 0 , which is not possible. Thus | β 0 | 1 . Similarly | γ 0 | 1 .

Case 2. X is a complex normed linear space.

Then | β 0 | = 1 implies that
1 = z 1 = | β 0 | x 0 + j = 1 n β j β 0 x α j > x 0 ,

unless β i = 0 i = 1 , 2 , , n .

Thus | β 0 | = 1 z 1 = β 0 x 0 z 1 = e i θ x 0 , similarly | γ 0 | = 1 z 2 = e i ϕ x 0 . Then x 0 = t e i θ x 0 + ( 1 t ) e i ϕ x 0 x 0 = z 1 = z 2 , which is not possible. Thus | β 0 | = 1 and | γ 0 | = 1 cannot hold simultaneously.

So, at least one of | β 0 | , | γ 0 | is less than 1.

Now x 0 = t z 1 + ( 1 t ) z 2 implies
t β 0 + ( 1 t ) γ 0 = 1 , t β j + ( 1 t ) γ j = 0 j = 1 , 2 , , n .
But | β 0 | < 1 or | γ 0 | < 1 implies
1 = | t β 0 + ( 1 t ) γ 0 | t | β 0 | + ( 1 t ) | γ 0 | < 1 ,

which is not possible.

Thus x 0 is an extreme point of B X . This completes the proof. □

The converse of the above theorem is, however, not always true. If x 0 is an extreme point of B X , then there may or may not exist a strongly orthonormal Hamel basis relative to x 0 in the sense of Birkhoff-James.

Example 2.2 (i) Consider ( R 2 , ) where the unit sphere S is given by S = { ( x , y ) R 2 : x = ± 1 and 1 y 1 } { ( x , y ) R 2 : x 2 2 y + y 2 = 0 and y 1 } { ( x , y ) R 2 : x 2 + 2 y + y 2 = 0 and y 1 }. Then ( 1 , 1 ) is an extreme point of the unit ball, but there exists no strongly orthonormal Hamel basis relative to ( 1 , 1 ) in the sense of Birkhoff-James.
  1. (ii)

    Consider ( R 2 , ) where the unit sphere S is given by S = { ( x , y ) R 2 : x = ± 1 and 1 y 1 } { ( x , y ) R 2 : x 2 + 2 y 3 = 0 and y 1 } { ( x , y ) R 2 : x 2 2 y 3 = 0 and y 1 }. Then ( 1 , 1 ) is an extreme point of the unit ball and { ( 1 , 1 ) , ( 1 , 1 ) } is a strongly orthonormal basis relative to ( 1 , 1 ) in the sense of Birkhoff-James.

     
  2. (iii)

    In l ( R 3 ) the extreme points of the unit ball are of the form ( ± 1 , ± 1 , ± 1 ) , and for the extreme point ( 1 , 1 , 1 ) , we can find a strongly orthonormal basis relative to ( 1 , 1 , 1 ) in the sense of Birkhoff-James which is { ( 1 , 1 , 1 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) } .

     

In the first two examples, the extreme point ( 1 , 1 ) is such that every neighborhood of ( 1 , 1 ) contains both extreme as well as non-extreme points, whereas in the third case the extreme point ( 1 , 1 , 1 ) is an isolated extreme point.

An element x in the boundary of a convex set S is called an exposed point of S iff there exists a hyperplane of support H to S through x such that H S = { x } . The notion of exposed points can be found in [5, 1315]. We next prove that if the extreme point x 0 is an exposed point of B X , then there exists a Hamel basis of X containing x 0 which is strongly orthonormal relative to x 0 in the sense of Birkhoff-James.

Theorem 2.3 Let X be a normed linear space and x 0 S X be an exposed point of B X . Then there exists a Hamel basis of X containing x 0 which is strongly orthonormal relative to x 0 in the sense of Birkhoff-James.

Proof As x 0 is an exposed point of B X , so there exists a hyperplane of support H to B X through x 0 such that H B X = { x 0 } . Then we can find a linear functional f on X such that H = { x X : f ( x ) = 1 } . Let H 0 = { x X : f ( x ) = 0 } . Then H 0 is a subspace of X. Let D = { x α : α Λ } be a Hamel basis of H 0 with x α = 1 . Clearly { x 0 } D is a Hamel basis of X. We claim that { x 0 } D is a strongly orthonormal set relative to x 0 in the sense of Birkhoff-James.

Consider a finite subset { x α 1 , x α 2 , , x α n 1 } of D and let ( λ 1 , λ 2 , , λ n 1 ) ( 0 , 0 , , 0 ) . Now if z = x 0 + j = 1 n 1 λ j x α j , then

So x 0 + j = 1 n 1 λ j x α j > 1 = x 0 . Thus { x 0 } D is a Hamel basis containing x 0 which is strongly orthonormal relative to x 0 in the sense of Birkhoff-James.

This completes the proof. □

We next prove the following theorem.

Theorem 2.4 Let X be a normed linear space and x 0 S X . If there exists a Hamel basis of X containing x 0 which is strongly orthonormal relative to x 0 in the sense of Birkhoff-James, then there exists a bounded invertible linear operator A on X such that A = A x 0 > A y for all y in S X with y λ x 0 , λ S K .

Proof Let { x 0 , x α : α Λ } be a Hamel basis of X which is strongly orthonormal relative to x 0 in the sense of Birkhoff-James.

Define a linear operator A on X by A ( x 0 ) = x 0 and A ( x α ) = 1 2 x α α Λ .

Clearly A is invertible. Take any z X such that z = 1 . Then z = λ 0 x 0 + j = 1 n 1 λ j x α j for some scalars λ j ’s and λ 0 .

If λ 0 = 0 , then A z = 1 2 z and so
A x 0 = 1 > 1 2 = A z .
If λ 0 0 , then as { x 0 , x α : α Λ } is a strongly orthonormal Hamel basis relative to x 0 in the sense of Birkhoff-James, so we get
1 = z = λ 0 x 0 + j = 1 n 1 λ j x α j | λ 0 | .
Hence we get
A z = λ 0 x 0 + 1 2 j = 1 n 1 λ j x α j = 1 2 ( λ 0 x 0 + j = 1 n 1 λ j x α j ) + 1 2 λ 0 x 0 1 2 z + 1 2 | λ 0 | 1 = A x 0 .

This proves that A 1 . Also A z = 1 iff | λ 0 | = 1 and λ j = 0 j = 1 , 2 , , n 1 .

Thus A z = 1 iff z = λ 0 x 0 with λ 0 S K . This completes the proof. □

We now prove the following theorem.

Theorem 2.5 Let X be a normed linear space and x 0 S X . If there exists a bounded linear operator A : X X which attains its norm only at the points of the form λ x 0 with λ S K , then x 0 is an exposed point of B X .

Proof Assume, without loss of generality, that A = 1 and by the Hahn-Banach theorem, there exists f S X such that f ( A ( x 0 ) ) = 1 . Clearly f o A = 1 as f S X and A = A x 0 = 1 . If y S X is such that | f o A ( y ) | = 1 , then A y = 1 .

Now A = 1 and A attains its norm only at the points of the form λ x 0 with λ S K , so y { λ x 0 : λ S K } .

Thus f o A attains its norm only at the points of the form λ x 0 with λ S K . Considering the hyperplane H = { x X : f o A ( x ) = 1 } , it is easy to verify that H B X = { x 0 } and so x 0 is an exposed point of B X . □

Thus we obtained complete characterizations of exposed points, which is stated clearly in the following theorem.

Theorem 2.6 For a normed linear space X and a point x S X , the following are equivalent:
  1. 1.

    x is an exposed point of B X .

     
  2. 2.

    There exists a Hamel basis of X containing x which is strongly orthonormal relative to x in the sense of Birkhoff-James.

     
  3. 3.

    There exists a bounded linear operator A on X which attains its norm only at the points of the form λx with λ S K .

     

We next give a characterization of a strictly convex space as follows.

Theorem 2.7 For a normed linear space X, the following are equivalent:
  1. 1.

    X is strictly convex.

     
  2. 2.

    For each x S X , there exists a Hamel basis of X containing x which is strongly orthonormal relative to x in the sense of Birkhoff-James.

     
  3. 3.

    For each x S X , there exists a bounded linear operator A on X which attains its norm only at the points of the form λx with λ S K .

     

Proof The proof follows from previous theorem and the fact that a normed linear space X is strictly convex iff every element of S X is an exposed point of B X . □

Remark 2.8 Even though the notions of strong Birkhoff-James orthogonality and Birkhoff-James orthogonality coincide in a Hilbert space, they do not characterize Hilbert spaces as ( R n , p ) ( 1 < p < , p 2 ) is not a Hilbert space, but the notions of strong Birkhoff-James orthogonality and Birkhoff-James orthogonality coincide there.

Declarations

Acknowledgements

We would like to thank the referee for their invaluable suggestion. We would also like to thank Professor TK Mukherjee for his invaluable suggestion while preparing this paper. The first author would like to thank Jadavpur University and DST, Govt. of India for the partial financial support provided through DST-PURSE project and the second author would like to thank UGC, Govt. of India for the financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Jadavpur University
(2)
Department of Mathematical Sciences, School of Science, Kathmandu University

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© Paul et al.; licensee Springer. 2013

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