Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 3, Article 79, 2005
ON AN ε-BIRKHOFF ORTHOGONALITY
JACEK CHMIELIŃSKI
I NSTYTUT M ATEMATYKI , A KADEMIA P EDAGOGICZNA W K RAKOWIE
P ODCHOR A̧ ŻYCH 2, 30-084 K RAKÓW, P OLAND
jacek@ap.krakow.pl
Received 18 February, 2005; accepted 28 July, 2005
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Birkhoff (Birkhoff-James) orthogonality, Approximate orthogonality, Semi-inner-product.
2000 Mathematics Subject Classification. 46B20, 46C50.
1. I NTRODUCTION
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]:
x⊥ B y ⇐⇒ ∀λ ∈ K : kx + λyk ≥ (1 − ε) kxk .
ε
We are going to give another definition of this concept.
(1.1)
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
The paper has been completed during author’s stay at the Silesian University in Katowice.
043-05
2
JACEK C HMIELI ŃSKI
2. B IRKHOFF A PPROXIMATE O RTHOGONALITY
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
kαx + λβyk2 = |α|2 x
+
λ
y
α β 2
2
≥ |α| kxk − 2ε kxk λ y α = kαxk2 − 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 .
√
Thus x⊥εD y ⇔ x⊥
y
with
ρ
=
ρ(ε)
=
1
−
1 − ε2 .
B
ρ
Then, for inner product spaces we have:
x⊥εD y
⇐⇒
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.
It is easily seen that x⊥B y ⇔ps(x, y) = 1. It is also apparent that x⊥εD 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
1
1
the maximum norm k(x1 , x2 )k := max{|x1 |, |x2 |}. Now, let x = (1, 0), y = 2 , 1 , ε = 2 . One
2
can verify that x⊥εB y (i.e., that max 1 + λ2 , |λ|
≥ 1 − |λ| holds for each λ ∈ R) but not
2
ε
ε
ε
x⊥D y (take λ = − 3 ). Thus ⊥B 6⊂ ⊥D .
√
2
On the other hand, for x = 1, 12 , y = (1, 0), ε = 23 we have max{|1 + λ|, 21 } ≥
√ 2
√
1 − 23 , i.e., x⊥εD y but not x⊥εB y (consider, for example, λ = 23 − 1). Thus ⊥εD 6⊂ ⊥εB .
See also Remark 4.1 for further comparison of ⊥εB and ⊥εD .
J. Inequal. Pure and Appl. Math., 6(3) Art. 79, 2005
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O N AN ε-B IRKHOFF O RTHOGONALITY
3
3. S EMI – INNER – PRODUCT ( APPROXIMATE ) O RTHOGONALITY
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;
(s2) [x|λy] = λ [x|y] , x, y ∈ X, λ ∈ K;
(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 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. [5, Theorem 3] or [4]). It
follows also in that case (see the proof of Theorem 3 in [5]):
(3.1)
lim
λ→0
λ∈R
kx + λyk − 1
= Re [y|x] , x, y ∈ S.
λ
Extending previous notations we define semi-orthogonality and approximate semi-orthogonality:
x⊥s y ⇔ [y|x] = 0;
x⊥εs y
⇔
| [y|x] | ≤ ε kxk · kyk ,
for x, y ∈ X and 0 ≤ ε < 1.
Obviously, for an inner–product space: ⊥s = ⊥ and ⊥εs = ⊥ε .
Proposition 3.1. For x, y ∈ X, if x⊥εs y, then x⊥εB y (i.e., ⊥εs ⊂ ⊥εB ).
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 whence
kx + λyk ≥ kxk + θε kyk · λ · eiϕ = kxk + θε kyk Re λeiϕ + iθε kyk Im λeiϕ .
J. Inequal. Pure and Appl. Math., 6(3) Art. 79, 2005
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JACEK C HMIELI ŃSKI
Therefore
kx + λyk2 ≥ kxk + θε kyk Re λeiϕ
2
+ θε kyk Im λeiϕ
iϕ
2
= kxk2 + 2θε kxk kyk Re λe
2
2 2
iϕ 2
iϕ 2
+ Im λe
+ θ ε kyk
Re λe
= 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 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|
whence
(3.2)
Re [λy|x + λy] + kx + λyk − 1 ≥ −2ε |λ| , for all λ ∈ K.
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 + n |λ0 | y − 1
λ0
|λ0 | λ0
≥ −2ε.
Re
y|x +
y +
|λ0 |
|λ0 |
n |λ0 |
n
Putting y 0 :=
inequality
λ0
y
|λ0 |
∈ S, ξn :=
|λ0 |
n
∈ R (ξn → 0 as n → ∞) we obtain from the above
kx + ξn y 0 k − 1
≥ −2ε.
ξn
Letting n → ∞, using continuity of the s.i.p. and (3.1)
Re [y 0 |x + ξn y 0 ] +
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.
J. Inequal. Pure and Appl. Math., 6(3) Art. 79, 2005
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O N AN ε-B IRKHOFF O RTHOGONALITY
5
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
1
Example 3.1. Consider the space l1 (with the norm kxk = ∞
i=1 |xi | for x = (x1 , x2 , . . .) ∈ l ).
Define
∞
X
xi yi
[x|y] := kyk
, x, y ∈ l1
|y
|
i
i=1
1
— a semi-inner-product in l . Let ε ∈ [0,
Then, for an arbitrary λ ∈ K:
√
yi 6=0
2 − 1) and let x = (1, 0, 0, . . .), y = (1, 1, ε, 0, . . .).
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,
1
[y|x] = 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].
4. S OME R EMARKS
Remark 4.1. 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) = 0 ⇔ ε = 0 and
η(ε)
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)
η(ε)
x⊥D y ⇒ x⊥εB y
for all ε ∈ [0, 1).
It follows from [3, Lemma 1] that for a closed, proper linear subspace G of a normed space
ε
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)
ε
G⊥D ⊂ G⊥B .
Therefore, we have
J. Inequal. Pure and Appl. Math., 6(3) Art. 79, 2005
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6
JACEK C HMIELI ŃSKI
Lemma 4.2. If X is a normed space with the s.i.p. [·|·] of the (APP)-type, then for an arbitrary
ε
proper and closed linear subspace G and an arbitrary ε ∈ [0, 1) the set G⊥B of all vectors
ε-Birkhoff orthogonal to G is nonzero.
We have also
Theorem 4.3. 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 .
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:
x⊥εB e ⇒ x⊥B e.
(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| .
Therefore
∀ λ ∈ K : (|x1 + λ| + 1 − x1 )2 ≥ 1 − 2ε |λ| .
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,
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 x =
3
kek
4
Then [x|e] =
λ = − 53 one has
3
4
3
4
3 1
, , 0, . . .
4 4
.
kxk, i.e, e⊥s x, whence (Proposition 3.1) e⊥B x. On the other hand, for
ke + λxk =
2
< 1 = kek,
3
i.e., ¬(e⊥B x).
J. Inequal. Pure and Appl. Math., 6(3) Art. 79, 2005
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O N AN ε-B IRKHOFF O RTHOGONALITY
7
R EFERENCES
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[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.
[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.
[10] T. SZOSTOK, On a generalization of the sine function, Glasnik Matematički, 38 (2003), 29–44.
J. Inequal. Pure and Appl. Math., 6(3) Art. 79, 2005
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