JJ II

(1)

volume 3, issue 4, article 51, 2002.

Received 30 March, 2002;

accepted 30 April, 2002.

Communicated by:S.S. Dragomir

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON MODULI OF EXPANSION OF THE DUALITY MAPPING OF SMOOTH BANACH SPACES

PAVLE M. MILI ˇCI ´C

Faculty of Mathematics University of Belgrade YU-11000, Yugoslavia.

EMail:pmilicic@hotmail.com

c

2000Victoria University ISSN (electronic): 1443-5756 050-02

(2)

On Moduli of Expansion of the Duality Mapping of Smooth

Banach Spaces Pavle M. Miliˇci´c

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of15

J. Ineq. Pure and Appl. Math. 3(4) Art. 55, 2002

http://jipam.vu.edu.au

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 spaceX.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 ofXandS(X)the unit sphere inX (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 mapJ : X → 2^X^∗ is called the dual map ifJ(0) = 0and forx ∈ X, x6= 0,

J(x) ={f ∈X^∗|f(x) =kfk kxk,kfk=kxk}.

The dual map of X^∗ into 2^X^∗∗ we denote by J^∗. The map τ is canonical linear isometry ofX intoX^∗∗.

It is well known that functional (1) g(x, y) := kxk

2

t→−0lim

kx+tyk − kxk

t + lim

t→+0

kx+tyk − kxk t

(3)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page3of15

always exists onX².IfXis(S),then (1) reduces to g(x, y) =kxklim

t→0

kx+tyk − kxk

t ;

the functionalgis linear in the second argument,J(x)is a singleton andg(x,·)∈ J(x).In this case we shall writeJ(x) =J x=f_x.Then[y, x] :=g(x, y),de- fines a so called semi-inner product[·,·](s.i.p) onX²which generates the norm of X, [x, x] =kxk²

, (see [1]). If X is an inner-product space (i.p. space) theng(x, y)is the usual i.p. of the vectorxand the vectory.

By the use of functionalg we define the angle between vectorxand vector y(x6= 0, y 6= 0)as

(2) cos (x, y) := g(x, y) +g(y, x) 2kxk kyk (see [3]). If(X,(·,·))is an i.p. space, then (2) reduces to

cos (x, y) = (x, y) kxk kyk.

We say thatX is a quasi-inner product space (q.i.p space) if the following equality holds

(3) kx+yk⁴− kx−yk⁴ = 8

kxk²g(x, y) +kyk²g(y, x)

, (x, y ∈X)¹

1If(·,·)is an i.p. onX² theng(x, y) = (x, y)and the equality (3) is the parallelogram equality.

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of15

The equality (3) holds in the spacel⁴,but does not hold in the space l¹.A q.i.p. spaceXis(SC)and(U S)(see [6] and [4]).

Alongside the modulus of convexity ofX, δ_X,and the modulus of smoothness ofX, ρ_X,defined by

δX(ε) = inf

1−

x+y 2

x, y ∈S(X) ; kx−yk ≥ε

; ρX(ε) = sup

1−

x+y 2

x, y ∈S(X) ; kx−yk ≤ε

;

we have defined in [5] the angle modulus of convexity ofX, δ_X⁰ , and the angle modulus of smoothness ofX, ρ⁰_X by:

δ⁰_X(ε) = inf

1−cos (x, y) 2

x, y ∈S(X) ; kx−yk ≥ε

; ρ⁰_X(ε) = sup

1−cos (x, y) 2

.

We also recall the known definition of modulus of flatness of X, η_X (Day’s modulus):

η_X(ε) = sup

2− kx+yk kx−yk

. We now quote three known results.

Lemma 1. (Theorem 6 in [7] and Theorem 6 in [1]). LetX be a real normed space which is (S),(SC)and(R).Then for allf ∈ X^∗ there exists a unique x∈X such that

f(y) = g(x, y), (y∈X).

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of15

Lemma 2. (Theorem 7 in [1]). LetX be a Banach space which is (U S) and (U C)and let[·,·]be an s.i.p. onX²which generates the norm onX (see [1]).

Then the dual spaceX^∗is(U S)and(U C)and the functional hJ x, J yi:= [y, x], (x, y ∈X), is an s.i.p on(X^∗)².

Lemma 3. (Proposition 3 in [2]). LetXbe a real normed space. Then forJ, J^∗ andτ on their respective domains we have

J⁻¹ =τ⁻¹J^∗ and J =J^∗−1τ.

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

hJ x, J yi=g(x, y) =g(f_y, f_x), (x, y ∈X).

Lemma 4. Let X be a real normed space which is(S), (SC)and(R). Then forx, y ∈S(X)we have

(4) 1−

x+y 2

≤ 1−cos (x, y)

2 ≤ kx−yk kfx−fyk

4 .

Proof. Under the hypothesis of Lemma4, using Lemma1, we havefx =g(x,·) (x∈X).Consequently,

kf_x−f_yk= sup{|g(x, t)−g(y, t)| |t∈S(X)}

≥g(x, t)−g(y, t) (t∈S(X)).

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of15

Fort= _kx−yk^x−y ,(x6=y),we obtain

(5) g

x, x−y kx−yk

−g

y, x−y kx−yk

≤ kfx−fyk.

SinceX is(S),the functionalg is linear in the second argument. Hence, from (5) we get

(6) 1−g(x, y)−g(y, x) + 1≤ kx−yk kf_x−f_yk. Using the inequality

1−

x+y 2

≤ 1−cos (x, y)

2 ≤ kx−yk

2

(see Lemma 1 in [5]) and the inequality (6) we obtain the inequality (4).

Lemma 5. LetX be a Banach space which is(U S)and(U C).Letδ_X^∗ be the modulus of convexity ofX^∗.Then for eachε > 0and for allx, y ∈ S(X)the following implications hold

kx−yk ≤2δX^∗(ε) =⇒ kfx−fyk ≤ε, (7)

kf_x−f_yk ≥ε=⇒ kx−yk ≥2δ_X^∗(ε). (8)

Proof. By Lemma2,X^∗ is a Banach space which is(U C)and(U S).SinceX^∗ is(U C),for eachε >0,we haveδ_X^∗(ε)>0and, for allx, y ∈S(X), (9) kfx+fyk>2−2δX^∗(ε) =⇒ kfx−fyk< ε.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of15

Under the hypothesis of Lemma5, by Remark1, we haveg(x, y) =g(f_y, f_x). 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(f_y, f_x)≤ kf_x+f_yk −1, so that we have

(11) kx−yk+kfx+fyk ≥2.

Now, letx, y ∈S(X)andkx−yk<2δ_X^∗(ε).Then, by (11) we obtain kf_x+f_yk>2−2δ_X^∗(ε).

Thus, by (9), we conclude that

(12) kx−yk<2δ_X^∗(ε) =⇒ kf_x−f_yk< ε.

On the other hand if kx−yk = 2δ_X^∗(ε) and kf_x−f_yk > ε, by (9), it follows

kx−yk+kf_x+f_yk ≤2.

So, by (11), we get

kx−yk+kf_x+f_yk= 2.

Hence, using (10), we conclude thatg(x, y) = 1− kx−yk,i.e.,g(x, x−y) = kxk kx−yk.Thus, sinceXis(SC),using Lemma 5 in [1], we getx=x−y, which is impossible. So, the implication (7) is correct. The implication (8) follows from the implication (12).

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of15

We now introduce a new definition.

According to the inequality (4), to make further progress in the estimates of the moduliδ_X, δ⁰_X, ρ_X, ρ⁰_X,it is convenient to introduce

Definition 1. LetX be(S)andx, y ∈S(X).The functione_J: [0,2]→[0,2], defined by

e_J(ε) := inf{kf_x−f_yk | kx−yk ≥ε}

will be called the lower modulus of expansion of the dual mappingJ.

The functione_J : [0,2]→[0,2],defined as

e_J(ε) := sup{kf_x−f_yk | kx−yk ≤ε}

is the upper modulus of expansion of the dual mappingJ.

Now, we quote our new results. Firstly, we note some elementary properties of the modulie_J ande_J.

Theorem 6. LetXbe(S). Then the following assertions are valid.

a) The functione_J is nondecreasing on[0,2]. b) The functioneJ is nondecreasing on[0,2]. c) e_J(ε)≤e_J(ε) (ε∈[0,2]).

d) IfX is a Hilbert space, thene_J(ε) =e_J(ε).

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of15

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 e_J(ε) >

e_J(ε).Then

inf{kf_x−f_yk | kx−yk=ε} ≥inf{kf_x−f_yk | kx−yk ≥ε}

>sup{kf_x−f_yk | kx−yk ≤ε}

≥sup{kf_x−f_yk | kx−yk=ε}, which is not possible.

d) In a Hilbert space, we have

kf_x−f_yk= sup{|(x, t)−(y, t)| |t ∈S(X)} ≤ kx−yk.

On the other hand, the functional f_x −f_y attains its maximum int = _kx−yk^x−y ∈ S(X).

Hencekx−yk = kf_x−f_yk. Because of that, we havee_J(ε) = e_J(ε) = ε.

In the next theorems some relation between moduliδ_X⁰ , ρ⁰_X,eJ, eJ are given.

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of15

Theorem 7. LetXbe(S),(SC)and(R).Then, forε ∈(0,2]we have a) δ⁰_X(ε)≤ 1

2e_J(ε) b) ρ⁰_X(ε)≤ ε

4e_J(ε), c) 2

ερ_X(ε)≤η_X(ε)≤ 1

2e_J(ε).

Proof. The proof of the assertions a) and b) follows immediately using the def- initions of the functionsδ⁰_X andρ⁰_X and the inequality (4).

c) Letx, y ∈S(X), x6=y.By Lemma4, we have 2− kx+yk

kx−yk = 2 kx−yk

1− kx+yk 2

≤ 1−cos (x, y) kx−yk

≤ kx−yk kf_x−f_yk 2kx−yk

= kfx−fyk

2 .

So 2− kx+yk

kx−yk ≤ kf_x−f_yk

2 .

Using the definition ofηX andeJ,we obtain η_X(ε)≤ 1

2e_J(ε).

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of15

On the other hand

(0<kx−yk ≤ε) =⇒

1

kx−yk ≥ 1 ε

=⇒ 2− kx+yk kx−yk ≥ 2

ε

1− kx+yk 2

. Because of that we have

η_X(ε)≥ 2

ερ_X(ε).

Remark 2. The last inequality is true for an arbitrary spaceX.

Corollary 8. For a q.i.p. space, it holds that

(13) e_J(ε)≥ε

2 4

(ε∈[0,2]).

Proof. By a) of Theorem 7and the inequality ₃₂^ε⁴ ≤ δ⁰_X(ε)(see Corollary 2 in [5]), we get (13).

Corollary 9. IfXis(S),(SC)and(R)then a) δ⁰_X∗(ε)≤ 1

2e_J(ε), b) ρ⁰_X∗ ≤ 1

2e_J^∗(ε),

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of15

c) 2

3ρX^∗(ε)≤ηX^∗(ε)≤ 1

2eJ^∗(ε).

Proof. It is well-known that if X is(S),(SC)and(R)thenX^∗ is (S),(SC) and(R).Hence Theorem7is valid forX^∗.

Theorem 10. LetX be a Banach space which is(U C)and(U S).Then, for all ε >0,we have the following estimations:

a) ρ⁰_X(2δ_X^∗(ε))≤ εδ_X^∗(ε)

2 ,

b) ρ⁰_X∗(2δ_X(ε))≤ εδ_X(ε) 2 , c) e_J^∗(ε)≥2δ_X^∗(ε),

d) e_J(2δ_X^∗(ε))≤ε, (e_J^∗(2δ_X(ε))≤ε).

Proof. a) Using, in succession, the definition of the functionρ⁰_X,the inequality (4) in Lemma2and the implication (7), we obtain:

ρ⁰_X(2δ_X^∗(ε)) = sup

1−cos (x, y) 2

kx−yk ≤2δ_X^∗(ε)

≤ 1

4sup{kx−yk kf_x−f_yk | kx−yk ≤2δ_X^∗(ε)}

≤ 1

42εδ_X^∗(ε)

= εδ_X^∗(ε)

2 .

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of15

b) If, in a), we setX^∗instead ofX (X^∗∗instead ofX^∗), we get (14) ρ⁰_X∗(2δ_X^∗∗(ε))≤ εδX^∗∗(ε)

2 .

LetF, G∈S(X^∗∗).Under the hypothesis of Theorem10, we have δ_X^∗∗(ε) = inf

1− kF +Gk 2

kF −Gk ≥ε

= inf

1− kτ x+τ yk 2

kτ x−τ yk ≥ε

= inf

1− kτ(x+y)k 2

kτ(x−y)k ≥ε

= inf

1− kx+yk 2

kx−yk ≥ε

=δ_X(ε).

Consequently the inequality (14) is equivalent to the inequality b).

c) Using, in succession, the definition of e_J, Lemma3, and the implication (8), we get

eJ^∗(ε) = inf{kJ^∗fx−J^∗fyk | kfx−fyk ≥ε}

= inf{kτ x−τ yk | kf_x−f_yk ≥ε}

≥2δ_X^∗(ε).

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of15

d) Using the definition ofe_J and the implication (7), we get

e_J(2δ_X^∗(ε)) = sup{kf_x−f_yk | kx−yk ≤2δ_X^∗(ε)} ≤ε.

Replacing, here,X^∗withX^∗∗andJwithJ^∗,we get the second inequality.

Since in a Banach spaceXwe have δX(ε)≤1−

r 1− ε²

4 and δX(ε)≤δ_X⁰ (ε) (see Theorem 1 in [5]), using b) and a) of Theorem10, we obtain Corollary 11. Under the hypothesis of Theorem10, we have

a) 2

ερ⁰_X∗(2δ_X(ε))≤δ_X(ε)≤ 2

εδ⁰_X(ε), b) ρ⁰_X(2δ_X^∗(ε))≤ ε

2 1− r

1− ε² 4

! .

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of15

References

[1] J.R. GILES, Classes of semi-inner product spaces, Trans. Amer. Math. Soc., 129 (1967), 436–446.

[2] C.R. De PRIMAANDW.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. MILI ˇCI ´C, Sur le g−angle dans un espace normé, Mat. Vesnik, 45 (1993), 43–48.

[4] P.M. MILI ˇCI ´C, A generalization of the parallelogram equality in normed spaces, J. Math. of Kyoto Univ., 38(1) (1998), 71–75.

[5] P.M. MILI ˇCI ´C, The angle modulus of the deformation of a normed space, Riv. Mat. Univ. Parma, (6) 3(2002), 101–111.

[6] P.M. MILI ˇCI ´C, On the quasi inner product spaces, Mat. Bilten, (Skopje), 22(XLVIII) (1998), 19–30.

[7] P.M. MILI ˇCI ´C, 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.