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volume 6, issue 3, article 79, 2005.

Received 18 February, 2005;

accepted 28 July, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

ON ANε-BIRKHOFF ORTHOGONALITY

JACEK CHMIELI ´NSKI

Instytut Matematyki

Akademia Pedagogiczna w Krakowie Podchor ¸a˙zych 2, 30-084 Kraków, Poland EMail:jacek@ap.krakow.pl

c

2000Victoria University ISSN (electronic): 1443-5756 043-05

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On anε-Birkhoff Orthogonality Jacek Chmieli ´nski

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J. Ineq. Pure and Appl. Math. 6(3) Art. 79, 2005

http://jipam.vu.edu.au

Abstract

We define an approximate Birkhoff orthogonality relation in a normed space.

We compare it with the one given by S.S. Dragomir and establish some properties of it. In particular, we show that in smooth spaces it is equivalent to the approximate orthogonality stemming from the semi-inner-product.

2000 Mathematics Subject Classification:46B20, 46C50

Key words: Birkhoff (Birkhoff-James) orthogonality, Approximate orthogonality, Semi-inner-product

The paper has been completed during author’s stay at the Silesian University in Katowice.

1. Introduction

In an inner product space, with the standard orthogonality relation⊥, one can consider the approximate orthogonality defined by:

x⊥^εy ⇔ | hx|yi | ≤εkxk kyk.

(|cos(x, y)| ≤εforx, y 6= 0).

The notion of orthogonality in an arbitrary normed space, with the norm not necessarily coming from an inner product, may be introduced in various ways.

One of the possibilities is the following definition introduced by Birkhoff [1]

(cf. also James [6]). LetXbe a normed space over the fieldK∈ {R,C}; then forx, y ∈X

x⊥By⇐⇒ ∀λ∈K:kx+λyk ≥ kxk.

We call the relation⊥B, a Birkhoff orthogonality (often called a Birkhoff-James orthogonality).

Our aim is to define an approximate Birkhoff orthogonality generalizing the

⊥^ε one. Such a definition was given in [3]:

(1.1) x⊥

ε^By⇐⇒ ∀λ ∈K:kx+λyk ≥(1−ε)kxk. We are going to give another definition of this concept.

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2. Birkhoff Approximate Orthogonality

Let us define an approximate Birkhoff orthogonality. Forε∈[0,1):

(2.1) x⊥^ε_By⇐⇒ ∀λ∈K:kx+λyk² ≥ kxk²−2εkxk kλyk. If the above holds, we say thatxisε-Birkhoff orthogonal toy.

Note, that the relation⊥^ε_B is homogeneous, i.e.,x⊥^ε_Byimpliesαx⊥^ε_Bβy (for arbitrary α, β ∈ K). Indeed, for any λ ∈ K we have (excluding the obvious caseα= 0)

kαx+λβyk² =|α|²

x+λβ αy

2

≥ |α|²

kxk²−2εkxk

λβ αy

=kαxk²−2εkαxkkλβyk.

Proposition 2.1. IfX is an inner product space then, for arbitraryε∈[0,1), x⊥^εy ⇐⇒ x⊥^ε_By.

We omit the proof – a more general result will be proved later (Theorem3.3).

As a corollary, forε= 0, we obtain the well known fact: x⊥By ⇔ x⊥y(in an inner product space).

Let us modify slightly the definition of Dragomir (1.1). Replacing1−εby

√1−ε² we obtain:

x⊥^ε_Dy ⇐⇒ ∀λ∈K: kx+λyk ≥√

1−ε²kxk.

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Thusx⊥^ε_Dy ⇔ x⊥_ρBywithρ=ρ(ε) = 1−√ 1−ε². Then, for inner product spaces we have:

x⊥^ε_Dy ⇐⇒ x⊥^εy (see [3, Proposition 1]).

T. Szostok [10], considering a generalization of the sine function introduced, for a real normed spaceX, the mapping:

s(x, y) =

( infλ∈R kx+λyk

kxk , forx∈X\ {0};

1, forx= 0.

It is easily seen that x⊥By ⇔ s(x, y) = 1. It is also apparent thatx⊥^ε_Dy ⇔ s(x, y) ≥ √

1−ε². Defining c(x, y) := ±p

1−s²(x, y)(generalized cosine) one getsx⊥^ε_Dy ⇔ |c(x, y)| ≤ε.

Let us compare the approximate orthogonalities ⊥^ε_D and ⊥^ε_B. In an inner product space both of them are equal toε-orthogonality⊥^ε. Thus one may ask if they are equal in an arbitrary normed space. This is not true. Moreover, neither ⊥^ε_B ⊂ ⊥^ε_D nor ⊥^ε_D ⊂ ⊥^ε_B holds generally (i.e., for an arbitrary normed space and all ε ∈ [0,1)). For, consider X = R² (over R) equipped with the maximum normk(x₁, x₂)k:= max{|x₁|,|x₂|}. Now, letx= (1,0),y= ¹₂,1

, ε = ¹₂. One can verify that x⊥^ε_By (i.e., that max

1 + ^λ₂

,|λ| ² ≥ 1− |λ|

holds for eachλ ∈R) but notx⊥^ε_Dy(takeλ =−²₃). Thus⊥^ε_B6⊂ ⊥^ε_D. On the other hand, for x = 1,¹₂

, y = (1,0), ε =

√ 3

2 we have max{|1 +λ|,¹₂}2

≥ 1−^√

3 2

2

, i.e.,x⊥^ε_Dybut not x⊥^ε_By (consider, for example,λ=

√3

2 −1). Thus⊥^ε_D 6⊂ ⊥^ε_B.

See also Remark1for further comparison of⊥^ε_B and⊥^ε_D.

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3. Semi–inner–product (approximate) Orthogonality

LetXbe a normed space overK∈ {R,C}. The norm inXneed not come from an inner product. However, (cf. G. Lumer [7] and J.R. Giles [5]) there exists a mapping[·|·] :X×X →Ksatisfying the following properties:

(s3) [x|x] =kxk², x∈X;

(s4) |[x|y]| ≤ kxk · kyk, x, y∈X.

(Cf. also [4].) We will call each mapping [·|·] satisfying (s1)–(s4) a semi- inner-product (s.i.p.) in a normed spaceX. Let us stress that we assume that a s.i.p. generates the given norm inX (i.e., (s3) is satisfied). Note, that there may exist infinitely many different semi-inner-products inX. There is a unique s.i.p.

inXif and only ifX is smooth (i.e., there is a unique supporting hyperplane at each point of the unit sphere S or, equivalently, the norm is Gâteaux differen- tiable onS – cf. [2, 4]). IfX is an inner product space, the only s.i.p. onX is the inner-product itself ([7, Theorem 3]).

We say that s.i.p. is continuous iffRe [y|x+λy] → Re [y|x]asR3 λ → 0 for allx, y ∈S. The continuity of s.i.p is equivalent to the smoothness ofX(cf.

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[5, Theorem 3] or [4]). It follows also in that case (see the proof of Theorem 3 in [5]):

(3.1) lim

λ→0λ∈R

kx+λyk −1

λ = Re [y|x], x, y ∈S.

Extending previous notations we define semi-orthogonality and approximate semi-orthogonality:

x⊥sy ⇔ [y|x] = 0;

x⊥^ε_sy ⇔ |[y|x]| ≤εkxk · kyk, forx, y ∈Xand0≤ε <1.

Obviously, for an inner–product space:⊥s =⊥and⊥^ε_s =⊥^ε. Proposition 3.1. Forx, y ∈X, ifx⊥^ε_sy, thenx⊥^ε_By(i.e.,⊥^ε_s ⊂ ⊥^ε_B).

Proof. Suppose thatx⊥^ε_sy, i.e.,|[y|x]| ≤εkxk · kyk. Then, for someθ∈[0,1]

and for someϕ∈[−π, π]we have:

[y|x] =θεkxk · kyk ·e^iϕ.

For arbitraryλ∈Kwe have:

kx+λyk · kxk ≥ |[x+λy|x]|

=

kxk² +λ[y|x]

=

kxk² +θεkxk · kyk ·λ·e^iϕ

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whence

kx+λyk ≥

kxk+θεkyk ·λ·e^iϕ

=

kxk+θεkykRe λe^iϕ

+iθεkykIm λe^iϕ . Therefore

kx+λyk² ≥ kxk+θεkykRe λe^iϕ2

+ θεkykIm λe^iϕ2

=kxk²+ 2θεkxk kykRe λe^iϕ +θ²ε²kyk²

Re λe^iϕ2

+ Im λe^iϕ2

=kxk²+ 2θεkxk kykRe λe^iϕ

+θ²ε²kλyk²

≥ kxk²+ 2θεkxk kykRe λe^iϕ

≥ kxk²+ 2θεkxk kyk − λe^iϕ

=kxk²−2θεkxk kλyk

≥ kxk²−2εkxk kλyk,

i.e.,x⊥^ε_By.

Since|[y|x]| ≤ kxk kyk, i.e.,x⊥¹_syfor arbitraryx, y, the above result gives alsox⊥¹_Byfor allx, y. That is the reason we restrictεto the interval[0,1).

Proposition 3.2. IfXis a continuous s.i.p. space andε∈[0,1), then⊥^ε_B ⊂ ⊥^ε_s. Proof. Suppose thatx⊥^ε_By. Because of the homogeneity of relations⊥^ε_Band⊥^ε_s we may assume, without loss of generality, that x, y ∈ S. Then, for arbitrary

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λ ∈Kwe have:

0≤ kx+λyk²−1 + 2ε|λ|= [x|x+λy] + [λy|x+λy]−1 + 2ε|λ|. Therefore

0≤Re [x|x+λy] + Re [λy|x+λy]−1 + 2ε|λ|

≤ |[x|x+λy]|+ Re [λy|x+λy]−1 + 2ε|λ|

≤ kx+λyk+ Re [λy|x+λy]−1 + 2ε|λ|

whence

(3.2) Re [λy|x+λy] +kx+λyk −1≥ −2ε|λ|, for allλ ∈K. Letλ₀ ∈K\ {0},n∈Nandλ= ^λ_n⁰. Then from (3.2) we have

Re λ₀

n y|x+ λ₀ n y

+

x+λ₀ ny

−1≥ −2ε|λ₀| n ;

Re λ₀

|λ₀|y|x+ |λ₀| n

λ₀

|λ₀|y

+

x+^|λ_n⁰^|_|λ^λ⁰

0|y −1

|λ0| n

≥ −2ε.

Puttingy⁰ := _|λ^λ⁰

0|y∈S,ξn := ^|λ_n⁰^| ∈R(ξn →0asn → ∞) we obtain from the above inequality

Re [y⁰|x+ξ_ny⁰] +kx+ξ_ny⁰k −1

ξ_n ≥ −2ε.

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Lettingn→ ∞, using continuity of the s.i.p. and (3.1) Re [y⁰|x] + Re [y⁰|x]≥ −2ε whence

Re [λ₀y|x]≥ −ε|λ₀|.

Putting−λ₀in the place ofλ₀ we obtainRe [λ₀y|x]≤ε|λ₀|whence

|Re [λ₀y|x]| ≤ε|λ₀| for arbitraryλ₀ ∈K. Now, takingλ₀ = [y|x]we get

Reh

[y|x]y|xi

≤ε|[y|x]|

whence|[y|x]|² ≤ε|[y|x]|and finally|[y|x]| ≤ε, i.e.,x⊥^ε_sy.

Without the additional continuity assumption, the inclusion ⊥^ε_B ⊂ ⊥^ε_s need not hold.

Example 3.1. Consider the space l¹ (with the normkxk = P∞

i=1|x_i|forx = (x₁, x₂, . . .)∈l¹). Define

[x|y] :=kyk

∞

X

i=1 yi6=0

x_iy_i

|y_i|, x, y ∈l¹

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— a semi-inner-product in l¹. Let ε ∈ [0,√

2−1) and let x = (1,0,0, . . .), y = (1,1, ε,0, . . .). Then, for an arbitraryλ∈K:

kx+λyk²− kxk²+ 2εkxk kλyk= (|1 +λ|+|λ|+|λε|)²−1 + 2ε(2 +ε)|λ|

≥(1 +|λ|ε)²−1 + 2ε(2 +ε)|λ|

= 2ε(3 +ε)|λ|+|λ|²ε²

≥0, i.e.,x⊥^ε_By(in fact,x⊥By). On the other hand,

[y|x] = 1 = 1

2 +εkxk kyk> εkxk kyk

whence¬(x⊥^ε_sy). In particular, for ε = 0, this shows that ⊥B 6⊂ ⊥s (cf. [4, 8, 9]).

From the last two propositions we have:

Theorem 3.3. IfX is a continuous s.i.p. space, then

⊥^ε_B =⊥^ε_s.

Moreover we obtain, forε= 0, (cf. [5, Theorem 2]) Corollary 3.4. IfX is a continuous s.i.p. space, then

⊥_B =⊥_s.

Conversely,⊥B ⊂ ⊥s implies continuity of s.i.p. (smoothness) – cf. [4] and [8].

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4. Some Remarks

Remark 1. Dragomir [3, Definition 5] introduces the following concept: The s.i.p.[·|·]is of (APP)-type if there exists a mappingη : [0,1)→[0,1)such that η(ε) = 0 ⇔ ε= 0andx⊥^η(ε)D yimpliesx⊥^ε_syfor allε ∈[0,1). It follows from Proposition3.1that in that case we have also

(4.1) x⊥^η(ε)_D y ⇒ x⊥^ε_By

for allε∈[0,1).

It follows from [3, Lemma 1] that for a closed, proper linear subspaceGof a normed space X and for an arbitrary ε ∈ (0,1), the set G⊥^ε_D of all vectors

⊥^ε_D-orthogonal toGis nonzero. Using (4.1) we get

(4.2) G⊥^η(ε)D ⊂G⊥^ε_B.

Therefore, we have

Lemma 4.1. IfX is a normed space with the s.i.p.[·|·]of the (APP)-type, then for an arbitrary proper and closed linear subspaceGand an arbitraryε∈[0,1) the setG⊥^ε_B of all vectorsε-Birkhoff orthogonal toGis nonzero.

We have also

Theorem 4.2. If X is a normed space with the s.i.p. [·|·] of the (APP)-type, then for an arbitrary closed linear subspace Gand an arbitraryε ∈ [0,1)the following decomposition holds:

X =G+G⊥^ε_B.

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Proof. FixGandε∈[0,1). It follows from [3, Theorem 3] that X =G+G⊥^η(ε)D .

Using (4.2) we get the assertion.

The final example shows that the set of allε-orthogonal vectors may be equal to the set of all orthogonal ones.

Example 4.1. Consider again the space l¹ with the s.i.p. defined above. Let e = (1,0, . . .). Observe that vectorsε-orthogonal toeare, in fact, orthogonal toe:

(4.3) x⊥^ε_Be ⇒ x⊥Be.

Indeed, let ε ∈ [0,1) be fixed and let x = (x₁, x₂, . . .) ∈ l¹ satisfy x⊥^ε_Be.

Because of the homogeneity of ⊥^ε_B we may assume, without loss of generality, thatkxk= 1andx₁ ≥0. Thus we have

∀λ∈K:kx+λek² ≥1−2ε|λ|.

Therefore

∀λ∈K: (|x₁+λ|+ 1−x₁)² ≥1−2ε|λ|.

Suppose thatx₁ >0. Takeλ∈Rsuch thatλ <0,λ >−x₁andλ >−2(1−ε).

Then we have

(x₁+λ+ 1−x₁)² ≥1 + 2ελ,

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which leads to λ ≤ −2(1 − ε) – a contradiction. Thus x₁ = 0, i.e, x = (0, x2, x3, . . .)and|x2|+|x3|+· · ·= 1. This yields, for arbitraryλ ∈K,

kx+λek=|λ|+ 1≥1 =kxk,

i.e.,x⊥Be. It follows from (4.3) that forG:= linewe have

G⊥^ε_B =G⊥B.

Note, that the implication e⊥^ε_Bx ⇒ e⊥Bx is not true. Take for example x = ³₄,¹₄,0, . . .

. Then[x|e] = ³₄kek kxk, i.e,e⊥³⁴sx, whence (Proposition3.1) e⊥

3 4

Bx. On the other hand, forλ=−⁵₃ one has ke+λxk= 2

3 <1 =kek, i.e.,¬(e⊥Bx).

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References

[1] G. BIRKHOFF, Orthogonality in linear metric spaces, Duke Math. J., 1 (1935), 169–172.

[2] M.M. DAY, Normed Linear Spaces, Springer-Verlag, Berlin – Heidelberg – New York, 1973.

[3] S.S. DRAGOMIR, On approximation of continuous linear functionals in normed linear spaces, An. Univ. Timi¸soara Ser. ¸Stiin¸t. Mat., 29 (1991), 51–

58.

[4] S.S. DRAGOMIR, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004.

[5] J.R. GILES, Classes of semi–inner–product spaces, Trans. Amer. Math.

Soc., 129 (1967), 436–446.

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

[7] G. LUMER, Semi–inner–product spaces, Trans. Amer. Math. Soc., 100 (1961), 29–43.

[8] J. RÄTZ, On orthogonally additive mappings III, Abh. Math. Sem. Univ.

Hamburg, 59 (1989), 23–33.

[9] J. RÄTZ, Comparison of inner products, Aequationes Math., 57 (1999), 312–321.

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[10] T. SZOSTOK, On a generalization of the sine function, Glasnik Matem- atiˇcki, 38 (2003), 29–44.