Journal of Inequalities in Pure and
Applied Mathematics
ON MODULI OF EXPANSION OF THE DUALITY MAPPING
OF SMOOTH BANACH SPACES
volume 3, issue 4, article 51,
2002.
PAVLE M. MILIČIĆ
Faculty of Mathematics
University of Belgrade
YU-11000, Yugoslavia.
EMail: pmilicic@hotmail.com
Received 30 March, 2002;
accepted 30 April, 2002.
Communicated by: S.S. Dragomir
Abstract
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Abstract
Let X be a Banach space which is uniformly convex and uniformly smooth.
We introduce the lower and upper moduli of expansion of the dual mapping
J of the space X. Some estimation of certain well-known moduli (convexity,
smoothness and flatness) and two new moduli introduced in [5] are described
with this new moduli of expansion.
Let (X, k·k) be a real normed space, X ∗ its conjugate space, X ∗∗ the second
conjugate of X and S (X) the unit sphere in X (S (X) = {x ∈ X| kxk = 1}) .
Moreover, we shall use the following definitions and notations.
The sign (S) denotes that X is smooth, (R) that X is reflexive, (U S) that
X is uniformly smooth, (SC) that X is strictly convex, and (U C) that X is
uniformly convex.
∗
The map J : X → 2X is called the dual map if J (0) = 0 and for x ∈ X,
x 6= 0,
J (x) = {f ∈ X ∗ |f (x) = kf k kxk , kf k = kxk} .
∗∗
The dual map of X ∗ into 2X we denote by J ∗ . The map τ is canonical
linear isometry of X into X ∗∗ .
It is well known that functional
kxk
kx + tyk − kxk
kx + tyk − kxk
lim
+ lim
(1) g (x, y) :=
t→−0
t→+0
2
t
t
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
Pavle M. Miličić
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always exists on X 2 . If X is (S) , then (1) reduces to
kx + tyk − kxk
;
t→0
t
g (x, y) = kxk lim
the functional g is linear in the second argument, J (x) is a singleton and g (x, ·) ∈
J (x) . In this case we shall write J (x) = Jx = fx . Then [y, x] := g (x, y) , defines a so called semi-inner
product [·, ·] (s.i.p) on X 2 which generates the norm
2
of X, [x, x] = kxk , (see [1]). If X is an inner-product space (i.p. space)
then g (x, y) is the usual i.p. of the vector x and the vector y.
By the use of functional g we define the angle between vector x and vector
y (x 6= 0, y 6= 0) as
cos (x, y) :=
(2)
g (x, y) + g (y, x)
2 kxk kyk
(see [3]). If (X, (·, ·)) is an i.p. space, then (2) reduces to
(x, y)
cos (x, y) =
.
kxk kyk
On Moduli of Expansion of the
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Banach Spaces
Pavle M. Miličić
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We say that X is a quasi-inner product space (q.i.p space) if the following
equality holds
(3) kx + yk4 − kx − yk4 = 8 kxk2 g (x, y) + kyk2 g (y, x) , (x, y ∈ X) 1
If (·, ·) is an i.p. on X 2 then g (x, y) = (x, y) and the equality (3) is the parallelogram
equality.
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The equality (3) holds in the space l4 , but does not hold in the space l1 . A
q.i.p. space X is (SC) and (U S) (see [6] and [4]).
Alongside the modulus of convexity of X, δX , and the modulus of smoothness of X, ρX , defined by
x + y δX (ε) = inf 1 − 2 x, y ∈ S (X) ; kx − yk ≥ ε ;
x + y ρX (ε) = sup 1 − x, y ∈ S (X) ; kx − yk ≤ ε ;
2 0
X, δX
,
and the angle
we have defined in [5] the angle modulus of convexity of
modulus of smoothness of X, ρ0X by:
1 − cos (x, y) 0
δX (ε) = inf
x, y ∈ S (X) ; kx − yk ≥ ε ;
2
1 − cos (x, y) 0
ρX (ε) = sup
x, y ∈ S (X) ; kx − yk ≤ ε .
2
We also recall the known definition
modulus):
2 − kx + yk
ηX (ε) = sup
kx − yk
of modulus of flatness of X, ηX (Day’s
x, y ∈ S (X) ; kx − yk ≤ ε .
We now quote three known results.
Lemma 1. (Theorem 6 in [7] and Theorem 6 in [1]). Let X be a real normed
space which is (S) , (SC) and (R) . Then for all f ∈ X ∗ there exists a unique
x ∈ X such that
f (y) = g (x, y) , (y ∈ X) .
On Moduli of Expansion of the
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Banach Spaces
Pavle M. Miličić
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Lemma 2. (Theorem 7 in [1]). Let X be a Banach space which is (U S) and
(U C) and let [·, ·] be an s.i.p. on X 2 which generates the norm on X (see [1]).
Then the dual space X ∗ is (U S) and (U C) and the functional
hJx, Jyi := [y, x] , (x, y ∈ X) ,
is an s.i.p on (X ∗ )2 .
Lemma 3. (Proposition 3 in [2]). Let X be a real normed space. Then for J, J ∗
and τ on their respective domains we have
J −1 = τ −1 J ∗ and J = J ∗−1 τ.
Remark 1. Under the hypothesis of Lemma 2, the mappings J, J ∗ and τ are
bijective mappings. Then, by Lemma 3, Lemma 2 and Lemma 1, in this case,
we have
hJx, Jyi = g (x, y) = g (fy , fx ) , (x, y ∈ X) .
Lemma 4. Let X be a real normed space which is (S) , (SC) and (R) . Then
for x, y ∈ S (X) we have
x + y 1 − cos (x, y)
kx − yk kfx − fy k
≤
(4)
1−
≤
.
2
2
4
On Moduli of Expansion of the
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Pavle M. Miličić
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Proof. Under the hypothesis of Lemma 4, using Lemma 1, we have fx = g (x, ·)
(x ∈ X) . Consequently,
kfx − fy k = sup {|g (x, t) − g (y, t)| | t ∈ S (X)}
≥ g (x, t) − g (y, t) (t ∈ S (X)) .
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For t =
x−y
,
kx−yk
(x 6= y) , we obtain
(5)
x−y
g x,
kx − yk
x−y
− g y,
kx − yk
≤ kfx − fy k .
Since X is (S) , the functional g is linear in the second argument. Hence, from
(5) we get
(6)
1 − g (x, y) − g (y, x) + 1 ≤ kx − yk kfx − fy k .
Using the inequality
x + y 1 − cos (x, y)
kx − yk
≤
1−
≤
2 2
2
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
Pavle M. Miličić
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(see Lemma 1 in [5]) and the inequality (6) we obtain the inequality (4).
Lemma 5. Let X be a Banach space which is (U S) and (U C) . Let δX ∗ be the
modulus of convexity of X ∗ . Then for each ε > 0 and for all x, y ∈ S (X) the
following implications hold
(7)
(8)
kx − yk ≤ 2δX ∗ (ε) =⇒ kfx − fy k ≤ ε,
kfx − fy k ≥ ε =⇒ kx − yk ≥ 2δX ∗ (ε) .
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Proof. By Lemma 2, X ∗ is a Banach space which is (U C) and (U S) . Since X ∗
is (U C) , for each ε > 0, we have δX ∗ (ε) > 0 and, for all x, y ∈ S (X) ,
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kfx + fy k > 2 − 2δX ∗ (ε) =⇒ kfx − fy k < ε.
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Under the hypothesis of Lemma 5, by Remark 1, we have g (x, y) = g (fy , fx ) .
Hence, by inequality
1 − kx − yk ≤ g (x, y) ≤ kx + yk − 1
(see Lemma 1 in [6]), we obtain
(10)
1 − kx − yk ≤ g (x, y) = g (fy , fx ) ≤ kfx + fy k − 1,
so that we have
(11)
kx − yk + kfx + fy k ≥ 2.
Now, let x, y ∈ S (X) and kx − yk < 2δX ∗ (ε) . Then, by (11) we obtain
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
Pavle M. Miličić
kfx + fy k > 2 − 2δX ∗ (ε) .
Thus, by (9), we conclude that
(12)
kx − yk < 2δX ∗ (ε) =⇒ kfx − fy k < ε.
On the other hand if kx − yk = 2δX ∗ (ε) and kfx − fy k > ε, by (9), it
follows
kx − yk + kfx + fy k ≤ 2.
So, by (11), we get
kx − yk + kfx + fy k = 2.
Hence, using (10), we conclude that g (x, y) = 1 − kx − yk , i.e., g (x, x − y) =
kxk kx − yk . Thus, since X is (SC) , using Lemma 5 in [1], we get x = x − y,
which is impossible. So, the implication (7) is correct. The implication (8)
follows from the implication (12).
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We now introduce a new definition.
According to the inequality (4), to make further progress in the estimates of
0
the moduli δX , δX
, ρX , ρ0X , it is convenient to introduce
Definition 1. Let X be (S) and x, y ∈ S (X) . The function eJ : [0, 2] → [0, 2] ,
defined by
eJ (ε) := inf {kfx − fy k | kx − yk ≥ ε}
will be called the lower modulus of expansion of the dual mapping J.
The function eJ : [0, 2] → [0, 2] , defined as
eJ (ε) := sup {kfx − fy k | kx − yk ≤ ε}
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
Pavle M. Miličić
is the upper modulus of expansion of the dual mapping J.
Now, we quote our new results. Firstly, we note some elementary properties
of the moduli eJ and eJ .
Theorem 6. Let X be (S). Then the following assertions are valid.
a) The function eJ is nondecreasing on [0, 2] .
b) The function eJ is nondecreasing on [0, 2] .
c) eJ (ε) ≤ eJ (ε) (ε ∈ [0, 2]) .
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d) If X is a Hilbert space, then eJ (ε) = eJ (ε) .
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Proof. The assertions a) and b) follow from the implications
ε1 < ε2 =⇒ {(x, y) | kx − yk ≥ ε1 } ⊃ {(x, y) | kx − yk ≥ ε2 }
(x, y ∈ S (X)) ,
ε1 < ε2 =⇒ {(x, y) | kx − yk ≤ ε1 } ⊂ {(x, y) | kx − yk ≤ ε2 }
(x, y ∈ S (X)) .
c) Assume, to the contrary, i.e., that there is an ε ∈ [0, 2] such that eJ (ε) >
eJ (ε) . Then
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
Pavle M. Miličić
inf {kfx − fy k | kx − yk = ε} ≥ inf {kfx − fy k | kx − yk ≥ ε}
> sup {kfx − fy k | kx − yk ≤ ε}
≥ sup {kfx − fy k | kx − yk = ε} ,
which is not possible.
d) In a Hilbert space, we have
kfx − fy k = sup {|(x, t) − (y, t)| | t ∈ S (X)} ≤ kx − yk .
x−y
On the other hand, the functional fx − fy attains its maximum in t = kx−yk
∈
S (X) .
Hence kx − yk = kfx − fy k . Because of that, we have eJ (ε) = eJ (ε) =
ε.
0
In the next theorems some relation between moduli δX
, ρ0X ,eJ , eJ are given.
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Theorem 7. Let X be (S) , (SC) and (R) . Then, for ε ∈ (0, 2] we have
1
0
a) δX
(ε) ≤ eJ (ε)
2
ε
b) ρ0X (ε) ≤ eJ (ε) ,
4
1
2
c) ρX (ε) ≤ ηX (ε) ≤ eJ (ε) .
ε
2
Proof. The proof of the assertions a) and b) follows immediately using the def0
initions of the functions δX
and ρ0X and the inequality (4).
c) Let x, y ∈ S (X) , x 6= y. By Lemma 4, we have
2 − kx + yk
2
kx + yk
=
1−
kx − yk
kx − yk
2
1 − cos (x, y)
≤
kx − yk
kx − yk kfx − fy k
≤
2 kx − yk
kfx − fy k
=
.
2
So
2 − kx + yk
kfx − fy k
≤
.
kx − yk
2
Using the definition of ηX and eJ , we obtain
1
ηX (ε) ≤ eJ (ε) .
2
On Moduli of Expansion of the
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Pavle M. Miličić
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On the other hand
1
1
(0 < kx − yk ≤ ε) =⇒
≥
kx − yk
ε
2 − kx + yk
2
kx + yk
=⇒
≥
1−
.
kx − yk
ε
2
Because of that we have
2
ηX (ε) ≥ ρX (ε) .
ε
On Moduli of Expansion of the
Duality Mapping of Smooth
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Remark 2. The last inequality is true for an arbitrary space X.
Corollary 8. For a q.i.p. space, it holds that
ε 4
(13)
eJ (ε) ≥
(ε ∈ [0, 2]) .
2
Proof. By a) of Theorem 7 and the inequality
[5]), we get (13).
Corollary 9. If X is (S) , (SC) and (R) then
ε4
32
0
≤ δX
(ε) (see Corollary 2 in
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1
0
eJ (ε) ,
a) δX
∗ (ε) ≤
2
b)
ρ0X ∗
1
≤ eJ ∗ (ε) ,
2
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c)
2
1
ρX ∗ (ε) ≤ ηX ∗ (ε) ≤ eJ ∗ (ε) .
3
2
Proof. It is well-known that if X is (S) , (SC) and (R) then X ∗ is (S) , (SC)
and (R) . Hence Theorem 7 is valid for X ∗ .
Theorem 10. Let X be a Banach space which is (U C) and (U S) . Then, for all
ε > 0, we have the following estimations:
a) ρ0X (2δX ∗ (ε)) ≤
εδX ∗ (ε)
,
2
b) ρ0X ∗ (2δX (ε)) ≤
εδX (ε)
,
2
On Moduli of Expansion of the
Duality Mapping of Smooth
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Pavle M. Miličić
c) eJ ∗ (ε) ≥ 2δX ∗ (ε) ,
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d) eJ (2δX ∗ (ε)) ≤ ε,
(eJ ∗ (2δX (ε)) ≤ ε) .
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ρ0X ,
Proof. a) Using, in succession, the definition of the function
the inequality (4) in Lemma 2 and the implication (7), we obtain:
1 − cos (x, y) 0
ρX (2δX ∗ (ε)) = sup
kx − yk ≤ 2δX ∗ (ε)
2
1
≤ sup {kx − yk kfx − fy k | kx − yk ≤ 2δX ∗ (ε)}
4
1
≤ 2εδX ∗ (ε)
4
εδX ∗ (ε)
=
.
2
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b) If, in a), we set X ∗ instead of X (X ∗∗ instead of X ∗ ), we get
(14)
ρ0X ∗ (2δX ∗∗ (ε)) ≤
εδX ∗∗ (ε)
.
2
Let F, G ∈ S (X ∗∗ ) . Under the hypothesis of Theorem 10, we have
kF + Gk δX ∗∗ (ε) = inf 1 −
kF − Gk ≥ ε
2
kτ x + τ yk = inf 1 −
kτ x − τ yk ≥ ε
2
kτ (x + y)k = inf 1 −
kτ (x − y)k ≥ ε
2
kx + yk = inf 1 −
kx − yk ≥ ε
2 = δX (ε) .
Consequently the inequality (14) is equivalent to the inequality b).
c) Using, in succession, the definition of eJ , Lemma 3, and the implication
(8), we get
∗
∗
eJ ∗ (ε) = inf {kJ fx − J fy k | kfx − fy k ≥ ε}
= inf {kτ x − τ yk | kfx − fy k ≥ ε}
≥ 2δX ∗ (ε) .
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
Pavle M. Miličić
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d) Using the definition of eJ and the implication (7), we get
eJ (2δX ∗ (ε)) = sup {kfx − fy k | kx − yk ≤ 2δX ∗ (ε)} ≤ ε.
Replacing, here, X ∗ with X ∗∗ and J with J ∗ , we get the second inequality.
Since in a Banach space X we have
r
ε2
0
δX (ε) ≤ 1 − 1 −
and δX (ε) ≤ δX
(ε)
4
On Moduli of Expansion of the
Duality Mapping of Smooth
Banach Spaces
(see Theorem 1 in [5]), using b) and a) of Theorem 10, we obtain
Pavle M. Miličić
Corollary 11. Under the hypothesis of Theorem 10, we have
2 0
2 0
ρX ∗ (2δX (ε)) ≤ δX (ε) ≤ δX
(ε) ,
ε
ε
!
r
2
ε
ε
.
1− 1−
b) ρ0X (2δX ∗ (ε)) ≤
4
2
a)
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References
[1] J.R. GILES, Classes of semi-inner product spaces, Trans. Amer. Math. Soc.,
129 (1967), 436–446.
[2] C.R. De PRIMA AND W.V. PETRYSHYN, Remarks on strict monotonicity
and surjectivity properties of duality mapping defined on real normed linear
spaces, Math. Z., 123 (1971), 49–55.
[3] P.M. MILIČIĆ, Sur le g−angle dans un espace normé, Mat. Vesnik, 45
(1993), 43–48.
[4] P.M. MILIČIĆ, A generalization of the parallelogram equality in normed
spaces, J. Math. of Kyoto Univ., 38(1) (1998), 71–75.
[5] P.M. MILIČIĆ, The angle modulus of the deformation of a normed space,
Riv. Mat. Univ. Parma, (6) 3(2002), 101–111.
[6] P.M. MILIČIĆ, On the quasi inner product spaces, Mat. Bilten, (Skopje),
22(XLVIII) (1998), 19–30.
[7] P.M. MILIČIĆ, Sur la géometrie d’un espace normé avec la proprieté (G),
Proceedings of the International Workshop in Analysis and its Applications,
Institut za Matematiku, Univ. Novi Sad (1991), 163–170.
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