Journal of Inequalities in Pure and
Applied Mathematics
ON AN ε-BIRKHOFF ORTHOGONALITY
JACEK CHMIELIŃSKI
Instytut Matematyki
Akademia Pedagogiczna w Krakowie
Podchora̧żych 2, 30-084 Kraków, Poland
volume 6, issue 3, article 79,
2005.
Received 18 February, 2005;
accepted 28 July, 2005.
Communicated by: S.S. Dragomir
EMail: jacek@ap.krakow.pl
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
043-05
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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.
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
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Contents
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Birkhoff Approximate Orthogonality . . . . . . . . . . . . . . . . . . . . 4
3
Semi–inner–product (approximate) Orthogonality . . . . . . . . . 6
4
Some Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
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1.
Introduction
In an inner product space, with the standard orthogonality relation ⊥, one can
consider the approximate orthogonality defined by:
x⊥ε y ⇔ | hx|yi | ≤ ε kxk kyk .
( | cos(x, y)| ≤ ε for x, 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]). Let X be a normed space over the field K ∈ {R, C}; then
for x, y ∈ X
x⊥B y ⇐⇒ ∀ 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]:
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
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(1.1)
x⊥ B y ⇐⇒ ∀λ ∈ K : kx + λyk ≥ (1 − ε) kxk .
ε
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. For ε ∈ [0, 1):
(2.1)
x⊥εB y ⇐⇒ ∀λ ∈ K : kx + λyk2 ≥ kxk2 − 2ε kxk kλyk .
If the above holds, we say that x is ε-Birkhoff orthogonal to y.
Note, that the relation ⊥εB is homogeneous, i.e., x⊥εB y implies αx⊥εB βy (for
arbitrary α, β ∈ K). Indeed, for any λ ∈ K we have (excluding the obvious
case α = 0)
2
β 2
2
kαx + λβyk = |α| x + λ y α β 2
2
≥ |α| kxk − 2ε kxk λ α y 2
= kαxk − 2εkαxkkλβyk.
Proposition 2.1. If X is an inner product space then, for arbitrary ε ∈ [0, 1),
x⊥ε y ⇐⇒ x⊥εB y.
We omit the proof – a more general result will be proved later (Theorem 3.3).
As a corollary, for ε = 0, we obtain the well known fact: x⊥B y ⇔ x⊥y (in an
inner product space).
√ Let us modify slightly the definition of Dragomir (1.1). Replacing 1 − ε by
1 − ε2 we obtain:
√
x⊥εD y ⇐⇒ ∀ λ ∈ K : kx + λyk ≥ 1 − ε2 kxk .
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
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Thus x⊥εD y ⇔ x⊥
y with ρ = ρ(ε) = 1 −
ρB
Then, for inner product spaces we have:
x⊥εD y
⇐⇒
√
1 − ε2 .
x⊥ε y
(see [3, Proposition 1]).
T. Szostok [10], considering a generalization of the sine function introduced,
for a real normed space X, the mapping:
(
inf λ∈R kx+λyk
, for x ∈ X \ {0};
kxk
s(x, y) =
1,
for x = 0.
ε
It is easily√
seen that x⊥B y ⇔ s(x, y) = 1.
pIt is also apparent that x⊥D y ⇔
s(x, y) ≥ 1 − ε2 . Defining c(x, y) := ± 1 − s2 (x, y) (generalized cosine)
one gets x⊥εD y ⇔ |c(x, y)| ≤ ε.
Let us compare the approximate orthogonalities ⊥εD and ⊥εB . In an inner
product space both of them are equal to ε-orthogonality ⊥ε . Thus one may ask
if they are equal in an arbitrary normed space. This is not true. Moreover,
neither ⊥εB ⊂ ⊥εD nor ⊥εD ⊂ ⊥εB holds generally (i.e., for an arbitrary normed
space and all ε ∈ [0, 1)). For, consider X = R2 (over R) equipped with the
maximum norm k(x1 , x2 )k := max{|x1 |, |x2 |}. Now, let x = (1, 0), y = 12 , 1 ,
2
ε = 21 . One can verify that x⊥εB y (i.e., that max 1 + λ2 , |λ|
≥ 1 − |λ|
2
ε
ε
ε
holds for each λ ∈ R) but not x⊥D y (take λ = − 3 ). Thus ⊥B 6⊂ ⊥D√.
On the other hand, for x = 1, 12 , y = (1, 0), ε = 23 we have
√ 2
2
1
max{|1 + λ|, 2 } ≥ 1 − 23 , i.e., x⊥εD y but not x⊥εB y (consider, for ex-
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
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√
ample, λ = 23 − 1). Thus ⊥εD 6⊂ ⊥εB .
See also Remark 1 for further comparison of ⊥εB and ⊥εD .
J. Ineq. Pure and Appl. Math. 6(3) Art. 79, 2005
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3.
Semi–inner–product (approximate)
Orthogonality
Let X be a normed space over K ∈ {R, C}. The norm in X need not come from
an inner product. However, (cf. G. Lumer [7] and J.R. Giles [5]) there exists a
mapping [·|·] : X × X → K satisfying the following properties:
(s1) [λx + µy|z] = λ [x|z] + µ [y|z] , x, y, z ∈ X, λ, µ ∈ K;
On an ε-Birkhoff Orthogonality
(s2) [x|λy] = λ [x|y] , x, y ∈ X, λ ∈ K;
Jacek Chmieliński
(s3) [x|x] = kxk2 , x ∈ X;
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(s4) | [x|y] | ≤ kxk · kyk , x, y ∈ X.
(Cf. also [4].) We will call each mapping [·|·] satisfying (s1)–(s4) a semiinner-product (s.i.p.) in a normed space X. Let us stress that we assume that a
s.i.p. generates the given norm in X (i.e., (s3) is satisfied). Note, that there may
exist infinitely many different semi-inner-products in X. There is a unique s.i.p.
in X if and only if X is smooth (i.e., there is a unique supporting hyperplane at
each point of the unit sphere S or, equivalently, the norm is Gâteaux differentiable on S – cf. [2, 4]). If X is an inner product space, the only s.i.p. on X is
the inner-product itself ([7, Theorem 3]).
We say that s.i.p. is continuous iff Re [y|x + λy] → Re [y|x] as R 3 λ → 0
for all x, y ∈ S. The continuity of s.i.p is equivalent to the smoothness of X (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)
kx + λyk − 1
= Re [y|x] , x, y ∈ S.
λ→0
λ
lim
λ∈R
Extending previous notations we define semi-orthogonality and approximate
semi-orthogonality:
x⊥s y ⇔ [y|x] = 0;
ε
x⊥s y
⇔
| [y|x] | ≤ ε kxk · kyk ,
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
for x, y ∈ X and 0 ≤ ε < 1.
Obviously, for an inner–product space: ⊥s = ⊥ and ⊥εs = ⊥ε .
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Proposition 3.1. For x, y ∈ X, if x⊥εs y, then x⊥εB y (i.e., ⊥εs ⊂ ⊥εB ).
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Proof. Suppose that x⊥εs y, i.e., | [y|x] | ≤ ε kxk · kyk. Then, for some θ ∈ [0, 1]
and for some ϕ ∈ [−π, π] we have:
[y|x] = θε kxk · kyk · eiϕ .
For arbitrary λ ∈ K we have:
kx + λyk · kxk ≥ | [x + λy|x] |
= kxk2 + λ [y|x]
= kxk2 + θε kxk · kyk · λ · eiϕ JJ
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whence
kx + λyk ≥ kxk + θε kyk · λ · eiϕ = kxk + θε kyk Re λeiϕ + iθε kyk Im λeiϕ .
Therefore
kx + λyk2 ≥ kxk + θε kyk Re λeiϕ
2
+ θε kyk Im λeiϕ
iϕ
2
= kxk2 + 2θε kxk kyk Re λe
2
2 + θ2 ε2 kyk2 Re λeiϕ
+ Im λeiϕ
= kxk2 + 2θε kxk kyk Re λeiϕ + θ2 ε2 kλyk2
≥ kxk2 + 2θε kxk kyk Re λeiϕ
≥ kxk2 + 2θε kxk kyk − λeiϕ = kxk2 − 2θε kxk kλyk
≥ kxk2 − 2ε kxk kλyk ,
i.e., x⊥εB y.
Since | [y|x] | ≤ kxk kyk, i.e., x⊥1s y for arbitrary x, y, the above result gives
also x⊥1B y for all x, y. That is the reason we restrict ε to the interval [0, 1).
Proposition 3.2. If X is a continuous s.i.p. space and ε ∈ [0, 1), then ⊥εB ⊂ ⊥εs .
Proof. Suppose that x⊥εB y. Because of the homogeneity of relations ⊥εB and ⊥εs
we may assume, without loss of generality, that x, y ∈ S. Then, for arbitrary
On an ε-Birkhoff Orthogonality
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λ ∈ K we 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|
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
whence
Re [λy|x + λy] + kx + λyk − 1 ≥ −2ε |λ| , for all λ ∈ K.
(3.2)
Let λ0 ∈ K \ {0}, n ∈ N and λ = λn0 . Then from (3.2) we have
λ0
λ0
λ0 |λ0 |
Re
y|x + y + x + y − 1 ≥ −2ε
;
n
n
n
n
|λ0 | λ0 x
+
y
−1
n |λ0 | λ0
|λ0 | λ0
y|x +
y +
Re
≥ −2ε.
|λ0 |
|λ0 |
n |λ0 |
n
0
λ0
y
|λ0 |
Putting y :=
above inequality
∈ S, ξn :=
|λ0 |
n
∈ R (ξn → 0 as n → ∞) we obtain from the
kx + ξn y 0 k − 1
Re [y |x + ξn y ] +
≥ −2ε.
ξn
0
0
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Letting n → ∞, using continuity of the s.i.p. and (3.1)
Re [y 0 |x] + Re [y 0 |x] ≥ −2ε
whence
Re [λ0 y|x] ≥ −ε|λ0 |.
Putting −λ0 in the place of λ0 we obtain Re [λ0 y|x] ≤ ε|λ0 | whence
|Re [λ0 y|x]| ≤ ε|λ0 | for arbitrary λ0 ∈ K.
Now, taking λ0 = [y|x] we get
h
i
Re [y|x]y|x ≤ ε| [y|x] |
whence | [y|x] |2 ≤ ε| [y|x] | and finally | [y|x] | ≤ ε, i.e., x⊥εs y.
Without the additional continuity assumption, the inclusion ⊥εB ⊂ ⊥εs need
not hold.
P
Example 3.1. Consider the space l1 (with the norm kxk = ∞
i=1 |xi | for x =
(x1 , x2 , . . .) ∈ l1 ). Define
∞
X
xi yi
[x|y] := kyk
, x, y ∈ l1
|y
|
i
i=1
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yi 6=0
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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 arbitrary λ ∈ K:
kx + λyk2 − kxk2 + 2ε kxk 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|2 ε2
≥ 0,
i.e., x⊥εB y (in fact, x⊥B y). On the other hand,
[y|x] = 1 =
1
kxk kyk > ε kxk kyk
2+ε
whence ¬(x⊥εs y). 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. If X is a continuous s.i.p. space, then
⊥εB = ⊥εs .
Moreover we obtain, for ε = 0, (cf. [5, Theorem 2])
Corollary 3.4. If X is a continuous s.i.p. space, then
⊥B = ⊥s .
Conversely, ⊥B ⊂ ⊥s implies continuity of s.i.p. (smoothness) – cf. [4] and
[8].
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
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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 mapping η : [0, 1) → [0, 1) such that
η(ε)
η(ε) = 0 ⇔ ε = 0 and x⊥D y implies x⊥εs y for all ε ∈ [0, 1). It follows from
Proposition 3.1 that in that case we have also
(4.1)
x⊥η(ε)
⇒ x⊥εB y
D y
for all ε ∈ [0, 1).
On an ε-Birkhoff Orthogonality
It follows from [3, Lemma 1] that for a closed, proper linear subspace G of
ε
a normed space X and for an arbitrary ε ∈ (0, 1), the set G⊥D of all vectors
⊥εD -orthogonal to G is nonzero. Using (4.1) we get
η(ε)
(4.2)
G⊥D
ε
⊂ G⊥B .
Therefore, we have
Lemma 4.1. If X is a normed space with the s.i.p. [·|·] of the (APP)-type, then
for an arbitrary proper and closed linear subspace G and an arbitrary ε ∈ [0, 1)
ε
the set G⊥B of all vectors ε-Birkhoff orthogonal to G is nonzero.
We have also
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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 G and an arbitrary ε ∈ [0, 1) the
following decomposition holds:
ε
X = G + G⊥B .
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Proof. Fix G and ε ∈ [0, 1). It follows from [3, Theorem 3] that
η(ε)
X = G + G⊥D .
Using (4.2) we get the assertion.
The final example shows that the set of all ε-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 vectors ε-orthogonal to e are, in fact, orthogonal
to e:
On an ε-Birkhoff Orthogonality
x⊥εB e ⇒ x⊥B e.
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(4.3)
Indeed, let ε ∈ [0, 1) be fixed and let x = (x1 , x2 , . . .) ∈ l1 satisfy x⊥εB e.
Because of the homogeneity of ⊥εB we may assume, without loss of generality,
that kxk = 1 and x1 ≥ 0. Thus we have
∀λ ∈ K : kx + λek2 ≥ 1 − 2ε |λ| .
Jacek Chmieliński
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Therefore
∀ λ ∈ K : (|x1 + λ| + 1 − x1 )2 ≥ 1 − 2ε |λ| .
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Suppose that x1 > 0. Take λ ∈ R such that λ < 0, λ > −x1 and λ > −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 arbitrary λ ∈ K,
kx + λek = |λ| + 1 ≥ 1 = kxk ,
i.e., x⊥B e. It follows from (4.3) that for G := lin e we have
ε
G⊥B = G⊥B .
Note, that the implication e⊥εB x ⇒ e⊥B x is not true. Take for example
3
x = 43 , 14 , 0, . . . . Then [x|e] = 34 kek kxk, i.e, e⊥4s x, whence (Proposition 3.1)
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
3
e⊥4B x. On the other hand, for λ = − 53 one has
ke + λxk =
i.e., ¬(e⊥B x).
2
< 1 = kek,
3
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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. Timişoara Ser. Ştiinţ. Mat., 29 (1991), 51–
58.
On an ε-Birkhoff Orthogonality
Jacek Chmieliński
[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.
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[7] G. LUMER, Semi–inner–product spaces, Trans. Amer. Math. Soc., 100
(1961), 29–43.
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[8] J. RÄTZ, On orthogonally additive mappings III, Abh. Math. Sem. Univ.
Hamburg, 59 (1989), 23–33.
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[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 Matematički, 38 (2003), 29–44.
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