generalized duality mapping

Journal of the Indian Math. Soc. Vol. 82, Nos. (1 - 2), (2015), 169–183.

GENERALIZED DUALITY MAPPING EWA SZLACHTOWSKA AND DOMINIK MIELCZAREK Abstract. In this paper we propose a definition of generalized duality mapping, in short g.d.m. This definition is closely related to the classical definition of normalized duality mapping. We explore some properties of g.d.m. such as continuity, surjectivity and injectivity. In the terms of g.d.m. we prove more general results concerning strict convexity.

(Received: November 7, 2013, Accepted: December 20, 2013) 1. Introduction Motivated by papers [2, 3, 4, 7, 11] we propose a definition of generalized duality mappings from a normed vector space to its dual space with a more general axiom system. G. Lumer [7], while trying to carry over a Hilbert space argument to a general Banach situation, used a suitable mapping from Banach space into its dual in order to make up for the lack of an inner-product. On a vector space he constructed a type of inner product with more general axiom system than that of Hilbert space. Fundamental properties and consequences of semi-inner product were explored by Giles [4]. A more general definition of semi-inner product was in [11]. The concept of semi-inner product has been proven useful both theoretically and practically. The application of semi-inner product in the theory of functional analysis was demonstrated, for example, in [5, 6, 8, 10, 11]. Our aim is to determine what further developments can be made for generalized duality mappings. It seems that our procedure should be useful in the study of, for example, of geometry of Banach space. The important fact is that a g.d.m. still provides one with sufficient structure to obtain certain nontrivial general results. In this paper we focus on studying properties of generalized duality mappings. We show that the continuity of 2010 Mathematics Subject Classification. 46B10,46B20, 46B25. Key words and phrases:Duality mapping, normalized duality mapping, semi-inner product, semi-inner product spaces. ISSN 0019–5839

c Indian Mathematical Society, 2015 .

169

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g.d.m. is equivalent to Gˆ ateaux differentiability of the norm. It is well know that Gˆ ateaux differentiability of the norm is equivalent to the smoothness of Banach space (see [9]). Moreover a relation, that we might call the orthogonality relation, can be defined by g.d.m. We also propose a more general definition of a strictly convex space. It is convenient to characterize strict convexity in terms of g.d.m. properties. For the concept of strict convexity we refer to [1] and references therein. 2. Generalized Duality Mapping G. Lumer [7] gave a definition of a normalized duality mapping JN : X → X ∗ defined by  JN (x) = x∗ ∈ X ∗ : hx, x∗ i = kxk2 and kx∗ k = kxk ,

where X ∗ is a dual space of X and h·, ·i is a duality pairing. We give a more general definition of a duality mapping. Definition 2.1. Let (X, k · k) be a Banach space and let k · k∗ : X → R+ be a function such that kxk∗ = 0

if and only if

x = 0.

∗

We say that a mapping J : X → 2X is a generalized duality mapping (g.d.m.) if it satisfies the following condition J(x) = {x∗ ∈ X ∗ :

hx, x∗ i = kxkkxk∗

and

kx∗ k = kxk∗ }

for

x ∈ X.

˜ Moreover we call a functional J˜: X → X ∗ a selection provided that J(x) ∈ J(x) for x ∈ X. kxk J(x) for x ∈ X \ {0}. It is evident that some Note that JN (x) = kxk ∗ properties of g.d.m. result from properties of JN , such as orthogonality and strict convexity. However, for the sake of completeness we present all the proofs, starting with a simple lemma.

Lemma 2.1. For every x ∈ X, J(x) is not an empty and convex subset of X ∗ . Proof. According to Hahn-Banach Theorem, for every x ∈ X there exists a functional y ∗ ∈ X ∗ , such that hx, y ∗ i = kxk

and ky ∗ k = 1.

Setting x∗ = kxk∗ y ∗ , we obtain hx, x∗ i = kxk∗ hx, y ∗ i = kxk∗ kxk and kx∗ k = kxk∗ , hence J(x) 6= ∅.

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Let x∗1 , x∗2 ∈ J(x) and let α ∈ [0, 1]. Then hx, αx∗1 + (1 − α)x∗2 i = α hx, x∗1 i + (1 − α) hx, x∗2 i = αkxkkxk∗ + (1 − α)kxkkxk∗ = kxkkxk∗ and kαx∗1 + (1 − α)x∗2 k ≤ αkx∗1 k + (1 − α)kx∗2 k = αkxk∗ + (1 − α)kxk∗ = kxk∗ . For x =

x kxk



we obtain

x , αx∗1 + (1 − α)x∗2 kxk



=α



x , x∗ kxk 1



+ (1 − α)



x , x∗ kxk 2



= αkxk∗ + (1 − α)kxk∗ = kxk∗ , hence kαx∗1 + (1 − α)x∗2 k = kxk∗ . This completes the proof.



A smooth space X is a normed space in which for any point x with kxk = 1 there exists a unique functional f ∈ X ∗ such that f (x) = kf k = 1. It is well known that a space X is smooth if and only if its norm has a Gˆateaux differential at all points x with kxk = 1. Lemma 2.2. The following conditions are equivalent: (1) The space X is smooth. (2) For every x0 ∈ X \ {0} the set J(x) is a singleton. (3) For every sequence {xα } ⊂ X such that kxα k = kx0 k = 1 xα x0 ˜ α ) → J(x ˜ 0 ) in σ(X ∗ , X)-topology, if → , then J(x kxα k∗ kx0 k∗ ˜ where J˜ : X → X ∗ is a functional such that J(x) ∈ J(x). Moreover ′ ∗ ˜ for any other selection J : X → X we have  ′  ˜ α ) → 0 in σ(X ∗ , X)-topology. J˜ (xα ) − J(x (4) For x0 ∈ X \ {0}, y ∈ X there exists lim

λ→0

kx0 + λyk − kx0 k . λ

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Proof. 1. ⇔ 2. It is obvious. 2. ⇒ 3. Let J(x0 ) = {x∗0 } and let {xα } ⊂ X be a sequence such that x0 xα ˜ α ) 6→ x∗0 . → , and J(x kxα k = kx0 k = 1, kxα k∗ kx0 k∗ Then for some M > 0 and each α we get ˜ α )k = kxα k∗ ≤ M. kJ(x ˜ α ) → x∗ ∈ X ∗ in Without loss of generality we can assume that J(x ∗ σ(X , X)-topology (otherwise we can take a subsequence, viewed as a net). Then       x0 x0 xα ∗ ∗ ˜ kx0 k∗ , x − 1 = kx0 k∗ , x − kxα k∗ , J(xα )         x0 xα x0 x0 ∗ ˜ ˜ ˜ ,x − , J(xα ) + , J(xα ) − , J(xα ) . ≤ kx0 k∗ kx0 k∗ kx0 k∗ kxα k∗ The right side of the above inequality tends to zero, therefore   x0 ∗ , x = 1. kx0 k∗

Hence hx0 , x∗ i = kx0 kkx0 k∗ .

(2.1)

Moreover D E ˜ α ) ≤ lim kxkkxα k∗ = kxkkx0 k∗ , |hx, x∗ i| = lim x, J(x α

α

∗

hence and from (2.1) it follows that kx k = kx0 k∗ and x∗ ∈ J(x0 ) = {x∗0 }. This leads to the contradiction because x∗ 6= x∗0 . ′ On the basis of the above considerations, for any selection J˜ it follows that ′ lim J˜ (xα ) = x∗0

α

in σ(X ∗ , X)-topology.

3. ⇒ 2. It is obvious. 1. ⇔ 4. It is obvious.



Definition 2.2. A normed vector space is Gˆ ateaux differentiable if for all x ∈ X \ {0}, y ∈ X there exists the limit lim

λ→0,λ∈R

kx + λyk − kxk . λ

Definition 2.3. A g.d.m. J is continuous if and only if there exists a selection J˜ such that D E D E ˜ ˜ + λy) Re y, J(x) Re y, J(x = (2.2) lim λ→0,λ∈R kx + λyk∗ kxk∗

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for x ∈ X \ {0}, y ∈ X . The selection J˜ is called a continuous selection. Definition 2.4. We say that a g.d.m. J is weakly continuous if and only if there exists a selection J˜ such that, for x ∈ X \ {0}, y ∈ X the following holds:

lim

λ→0,λ∈R

D E ˜ + λy) Re x, J(x kx + λyk∗

= kxk.

(2.3)

The selection J˜ is called a weakly continuous selection. Theorem 2.1. If a g.d.m. J is continuous, then it is weakly continuous. Proof. For x ∈ X \ {0}, y ∈ X it follows: D E Re x, J˜(x + λy) kx + λyk∗

=

D E ˜ + λy) Re x + λy − λy, J(x kx + λyk∗

D

E ˜ + λy) Re x + λy, J(x

D E ˜ + λy) Re y, J(x

−λ kx + λyk∗ kx + λyk∗ D E ˜ + λy) Re y, J(x = kx + λyk − λ . kx + λyk∗

=

Hence, (2.3) follows after passage to the limit as λ → 0, λ ∈ R.



Theorem 2.2. A g.d.m. J is a continuous if and only if the norm k · k is Gˆ ateaux differentiable. Proof. For x ∈ X \ {0}, y ∈ X and λ > 0 we show that (2.2) holds for any selection J˜.

kx + λyk − kxk kx + λykkxk∗ − kxkkxk∗ = λ λkxk∗ D E E D ˜ x + λy, J(x) − x, J˜(x) ≥ λkxk∗ D E D E ˜ ˜ Re x + λy, J(x) − x, J(x) ≥ λkxk∗ D E ˜ Re y, J(x) = kxk∗

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and kx + λyk − kxk kx + λykkx + λyk∗ − kxkkx + λyk∗ = λ λkx + λyk∗ D E D E ˜ + λy) − x, J(x ˜ + λy) x + λy, J(x ≤ λkx + λyk∗ D E D E ˜ + λy) − Re x, J˜(x + λy) x + λy, J(x ≤ λkx + λyk∗ D E ˜ + λy) Re y, J(x . = kx + λyk∗ Hence, for λ > 0, we have D E D E ˜ ˜ + λy) Re y, J(x) Re y, J(x kx + λyk − kxk ≤ . ≤ kxk∗ λ kx + λyk∗

(2.4)

Analogously, for λ < 0 D E ˜ Re y, J(x) kxk∗

D E ˜ + λy) Re y, J(x kx + λyk − kxk ≥ . ≥ λ kx + λyk∗

kx + λyk − kxk , then there λ D E ˜ + λy) Re y, J(x

Therefore, by (2.4), (2.5) if there exists exists lim

(2.5)

lim

λ→0,λ∈R

kx + λyk∗ E ˜ Re y, J(x) and these limits are equal to . kxk∗ ˜ Then the Conversely, let us assume that (2.2) is true for any selection J. ˜ equations (2.4), (2.5) hold for an selection J, which completes the proof.  λ→0,λ∈R

D

Note that if the norm k · k is Gˆateaux differentiable, then (2.2) is satisfied ˜ Moreover by (2.4) we have for every selection J. D E ˜ + λy) Re y, J(x kx + 2λyk − kx + λyk ≤ kx + λyk∗ λ kx + 2λyk − kxk kx + λyk − kxk =2 − , 2λ λ for λ > 0. Analogously, for λ < 0. Hence, based on Theorem 2.2 one can easily obtain the following corollary:

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Corollary 2.1. For every selection J˜ and x ∈ X \ {0}, y ∈ X we have: D E ˜ + λy) Re y, J(x kx + λyk − kxk , = lim+ 1. lim+ λ kx + λyk∗ λ→0 λ→0 D E ˜ + λy) Re y, J(x kx + λyk − kxk 2. lim , = lim λ kx + λyk∗ λ→0− λ→0− D E ˜ Re y, J(x) kx + λyk − kxk , ≥ 3. lim λ kxk∗ λ→0+ D E ˜ Re y, J(x) kx + λyk − kxk 4. lim− . ≤ λ kxk∗ λ→0 Definition 2.5. We say that k · k∗ is smooth if for each x ∈ X \ {0}, y ∈ X there exists a finite limit lim

λ→0,λ∈R

kx + λyk∗ − kxk∗ . λ

We denote right-handed and left-handed derivatives by kx + λyk∗ − kxk∗ , λ kx + λyk∗ − kxk∗ . kxk− ∗ (y) = lim− λ λ→0 kxk+ ∗ (y) = lim

λ→0+

Lemma 2.3. Let X be a Banach space. If k · k∗ is smooth, then lim

λ→0,λ∈R

kx + λyk∗ = kxk∗

for any x ∈ X \ {0}, y ∈ X and D E D E ˜ + λy) − x, J(x) ˜ Re x, J(x ′ lim = kxkkxk∗ (y) λ→0,λ∈R λ ˜ for any selection J. Proof. It is easy to see that if k · k∗ is smooth, then it is continuous, i.e. lim

λ→0,λ∈R

kx + λyk∗ = kxk∗ .

For any selection J˜ we obtain kx + λykkx + λyk∗ − kxkkxk∗ λ kx + λykkx + λyk∗ − kx + λykkxk∗ + kx + λykkxk∗ − kxkkxk∗ = λ kx + λyk − kxk kx + λyk∗ − kxk∗ + kxk∗ , = kx + λyk λ λ

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or kx + λykkx + λyk∗ − kxkkxk∗ λ D E D E ˜ + λy) − x, J(x) ˜ x + λy, J(x = λ D E D E ˜ ˜ D E Re x, J(x + λy) − x, J(x) ˜ + λy) = + Re y, J(x λE D D E D E ˜ + λy) ˜ ˜ Re y, J(x Re x, J(x + λy) − x, J(x) . + kx + λyk∗ = λ kx + λyk∗ If there exists a right-handed derivative of k · k∗ , then by the convexity of the norm and Corollary 2.1 there exists a limit D E D E ˜ + λy) − x, J˜(x) Re x, J(x lim λ λ→0+ and it is equal to kxkkxk+ ∗ (y). Analogously, if there exists a left-handed derivative of k · k∗ , then D E D E Re x, J˜(x + λy) − x, J˜(x) lim = kxkkxk− ∗ (y), λ λ→0− which finishes the proof.



Lemma 2.4. Let J˜ be a selection such that for any x ∈ X \ {0}, y ∈ X there exists a limit D E D E ˜ + λy) − x, J(x) ˜ Re x, J(x lim λ→0,λ∈R λ and lim

λ→0,λ∈R

kx + λyk∗ = kxk∗ ,

then k · k∗ is smooth. Moreover D E D E ˜ + λy) − x, J˜(x) Re x, J(x ′ lim = kxkkxk∗ (y). λ→0,λ∈R λ Proof. By computations analogous to those in the proof of Lemma 2.3 we obtain D E D E ˜ + λy) − x, J(x) ˜ Re x, J(x = kxkkxk+ lim ∗ (y) λ λ→0+ and D E D E Re x, J˜(x + λy) − x, J˜(x) lim = kxkkxk− ∗ (y), λ λ→0−

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hence − kxkkxk+ ∗ (y) = kxkkxk∗ (y),

which completes the proof.



2.1. Strictly Convex Spaces. We give a definition of strictly convex spaces in terms of g.d.m.. This definition is equivalent to the classical concept of strict convexity (see [1]). Definition 2.6. LetEJ˜ be an arbitrary selection. A space X is strictly convex D ˜ if whenever x, J(y) = kxkkyk∗ , where x, y 6= 0, then y = λx for some real λ > 0. Theorem 2.3. A space X is a strictly convex space if and only if whenever kx + yk = kxk + kyk, where x, y 6= 0, then y = λx for some real λ > 0. Proof. Assuming that X is a strictly convex space and kx + yk = kxk + kyk. Then for any selection J˜ it follows: D E ˜ + y) = kx + ykkx + yk∗ = kxkkx + yk∗ + kykkx + yk∗ x + y, J(x D E D E ˜ + y) ≥ Re x, J˜(x + y) + Re y, J(x D E D E ˜ + y) = x + y, J(x ˜ + y) . = Re x + y, J(x Hence, D E D E ˜ + y) = x, J(x ˜ + y) = kxkkx + yk∗ , Re x, J(x D E D E ˜ + y) = y, J(x ˜ + y) = kykkx + yk∗ . Re y, J(x Therefore there exist α, β > 0 such that x + y = αx and x + y = βy, thus y = α/βx. D E ˜ Conversly, let J˜ be a selection such that x, J(y) = kxkkyk∗. Then D E D E D E ˜ ˜ kx + ykkyk∗ ≥ x + y, J(y) = x, J˜(y) + y, J(y)

= kxkkyk∗ + kykkyk∗ = (kxk + kyk)kyk∗ ≥ kx + ykkyk∗.

Hence, kx + yk = kxk + kyk, and therefore there exists α > 0 such that y = αx, which completes the proof.  A function k · k∗ has considerable structure when it possesses additional properties.

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Definition 2.7. The function k·k∗ satisfies Condition A, if whenever kxk∗ = kyk∗, then kxk = kyk. We say that the function k · k∗ satisfies Condition B, if whenever kxk = kyk, then kxk∗ = kyk∗. Lemma 2.5.D Let X Ebe a Dstrictly E convex space. Then, if there exists a selection ˜ ˜ ˜ J such that x, J (y) = x, J(z) for all x ∈ X, then there exists α > 0 such

that y = αz. Moreover, if the function k · k∗ satisfies Condition A, then y = z. D E D E ˜ Proof. Let J˜ be a selection such that x, J(y) = x, J˜(z) for all x ∈ X. Setting x = y it follows: D E D E ˜ ˜ kykkyk∗ = y, J(y) = y, J(z) ≤ kykkzk∗.

Hence, D kyk∗ E≤ kzk∗ and analogously, kzk∗ ≤ kyk∗ . Therefore, kyk∗ = kzk∗ ˜ and y, J(z) = kykkzk∗. Hence, by strict convexity there exists α > 0 such that y = αz. If the function k · k∗ satisfies Condition A, then kyk = kzk. Thus y = z.  Definition 2.8. A g.d.m. J is injective, i.e. if x 6= y, then J(x) ∩ J(y) = ∅.

Now we specify the correlation between strict convexity and injectivity of mapping J. Lemma 2.6. Let X be a strictly convex Banach space. If k · k∗ satisfies Condition A, then g.d.m. J is injective. Proof. Let us assume that there exist x, y, x 6= y such that J(x) ∩ J(y) 6= ∅. Then there is z ∗ ∈ J(x) ∩ J(y) such that kz ∗ k = kxk∗ = kyk∗ , hx, z ∗ i = kxkkxk∗

and

hy, z ∗ i = kykkyk∗.

Therefore for any selection J˜ we obtain D E ˜ x, J(y) = hx, z ∗ i = kxkkxk∗ = kxkkyk∗. By strict convexity of the space X there exists α > 0 such that y = αx. Moreover, kxk∗ = kyk∗ . From Condition A, we have kxk = kyk, hence α = 1, which is a contradiction.  Lemma 2.7. Let X be a Banach space. If k · k∗ satisfies Condition B and g.d.m. J is injective, then X is a strictly convex space. D E ˜ Proof. Assuming that there exists a selection J˜ such that x, J(y) = kxkkyk∗ for x, y ∈ X, then   kyk D ˜ E kyk kyk ˜ x, J (y) = x, J(y) = kxkkyk∗ = kykkyk∗ kxk kxk kxk

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and from Condition B it follows that

kyk

kyk∗ = x . kxk ∗





kyk kxk x

˜ Therefore, J(y) ∈J . From injectivity of J it follows that y = which completes the proof.

kyk kxk x,



Example 2.1. Suppose there exists function ϕ : R+ → R+ such that kxk∗ = ϕ (kxk) ,

(2.6)

for x ∈ X. It is obvious that in this case k·k∗ satisfies Condition B. Additionally, if ϕ is injection then k · k∗ fulfills Condition A. Moreover, if ϕ(t) = t1/p , p ∈ (1, +∞), then we obtain a semi-inner product of type p (see [11]). 2.2. Orthogonality Relation. We define an orthogonality relation and show that if g.d.m. is continuous, it is equivalent to an orthogonality in the sense of Birkhoff. Definition 2.9. We say that D x is normal E to y and y is transversal to x if there ˜ ˜ exists a selection J such that y, J(x) = 0.

For a normed space we can also define an orthogonality in the sense of Birkhoff. Definition 2.10. A vector x is orthogonal to y (in the sense of Birkhoff ) if kx + λyk ≥ kxk

for all

λ ∈ C.

for all

λ ∈ C.

Lemma 2.8. If x is normal to y, then kx + λyk ≥ kxk

Proof. Let x be normal to y. Thus, D E ˜ kx + λykkxk∗ ≥ x + λy, J(x) = D E D E ˜ x, J˜(x) + λ y, J(x) = kxkkxk∗ .

Therefore, ||x + λy|| ≥ ||x|| for all complex λ.



Lemma 2.9. If x is orthogonal to y in the sense of Birkhoff and J is continuous, then x is normal to y. Proof. Let J˜ be a continuous selection. If kx + λyk ≥ kxk for all λ ∈ C, then 0 ≤ kx + λyk∗ kx + λyk − kx + λyk∗ kxk D E D E ˜ + λy) − x, J˜(x + λy) ≤ x + λy, J(x D E n D Eo D E ˜ + λy) + Re λ y, J(x ˜ + λy) ˜ + λy) . ≤ Re x, J(x − Re x, J(x

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n D Eo ˜ + λy) Therefore, Re λ y, J(x ≥ 0. For real λ we have D E ˜ + λy) ≥ 0 for λ ≥ 0; Re y, J(x D E ˜ + λy) ≤ 0 for λ ≤ 0. Re y, J(x From the continuity condition, for real λ, we have D E D E ˜ + λy) ˜ Re y, J(x Re y, J(x) lim = λ→0 kx + λyk∗ kxk∗ + − through positive D Evalues for λ → 0 and through negative values for λ → 0 . ˜ Thus Re y, J(x) = 0. For imaginary λ, say λ = iλ1 with λ1 real, n D Eo D E ˜ + λy) ˜ + λ1 iy) ≥ 0 Re λ y, J(x = λ1 Re iy, J(x

D E ˜ and from the continuity condition we obtain Re iy, J(x) = 0, D E ˜ i.e. Im y, J(x) = 0. D E ˜ Therefore, y, J(x) = 0.



2.3. Reflexive Space. We describe how the surjectivity of J relates to the reflexivity of the space X. First we give the definition of the surjectivity of J. Definition 2.11. The mapping J is surjective if and only if the following condition holds: [ J(x) = X ∗ . x∈X

Theorem 2.4. If the mapping J is surjective, then X is a reflexive space and for every y ∗ ∈ X ∗ there exists z ∈ X such that kz + λwk ≥ kzk for w ∈ ker(y ∗ ) and λ ∈ C. Proof. (i) reflexivity. The mapping J is surjective, i.e. for every y ∗ ∈ X ∗ there exists x ∈ X such that y ∗ ∈ J(x). Hence, hx, y ∗ i = kxkkxk∗ = kxkky ∗ k. According to James-Mazur Theorem, the space X is reflexive.

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(ii) Let y ∗ ∈ X ∗ . Then there exists z such that y ∗ ∈ J(z) and hz, y ∗ i = kzkkzk∗

and ky ∗ k = kzk∗.

Let w ∈ ker(y ∗ ). Then for λ ∈ C it follows: kz + λwkkzk∗ ≥ hz + λw, y ∗ i = hz, y ∗ i = kzkkzk∗. Thus, kz + λwk ≥ kzk ∗

for w ∈ ker(y ) and λ ∈ C.



Theorem 2.5. If for every y ∗ ∈ X ∗ there exists z ∈ X such that for w ∈ ker(y ∗ ) and λ ∈ C kz + λwk ≥ kzk and kzk∗ = ky ∗ k and hz, y ∗ i ∈ R+ , then J is a surjection. Proof. We will show that hz, y ∗ i = kzkkzk∗ = kzkky ∗k, hence it follows that y ∗ ∈ J(z). Let z ∈ X be such that for w ∈ ker(y ∗ ) and λ ∈ C kz + λwk ≥ kzk and kzk∗ = ky ∗ k. Therefore, kzk = inf {kz − yk : y ∈ ker(y ∗ )} . Hence z 6∈ ker(y ∗ ). ∗ Thus, X = span(z) ⊕ ker(y ∗ ). Let g = kyy∗ k . Then, for every y ∈ X there exists α ∈ C and w ∈ ker(y ∗ ) such that y = αz +w. Moreover, there exists a sequence {yn }∞ n=1 such that kyn k = 1 and |g(yn )| → kgk = 1 as n → ∞. Without loss of generality it follows that g(yn ) → 1 and g(yn ) = g(αn z + wn ) = g(αn z) ≤ kαn zk = |αn |kzk



1

≤ |αn | z + wn = kyn k. αn

Hence, g(αz) = |α|kzk = 1 and g(z) = kzk. Therefore, hz, y ∗ i = kzkkzk∗, which finishes the proof.



Let us assume that there exists a function ϕ : R+ → R+ and norm k · k1 on X such that 1. ϕ(t) = 0 if and only if t = 0.

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2. kxk∗ = ϕ (kxk1 ) for x ∈ X. Then we prove the following: Theorem 2.6. A g.d.m. J is surjective if and only if X is a reflexive and a mapping ϕ is surjective. Proof. Let us suppose that J is surjective. Therefore, it follows that X is reflexive and for y ∗ ∈ X ∗ there exists z ∈ X such that y ∗ ∈ J(z). Then we obtain the following equation: ky ∗ k = kzk∗ = ϕ (kzk1) . Hence, ϕ is surjective. Conversely, let us suppose that ϕ is surjective and X is reflexive. Since ϕ is surjective, there exists f : R+ → R+ such that ϕ (f (t)) = t for t ∈ R+ . Let y ∗ ∈ X ∗ . Then, from Mazur theorem it follows that there exists z in X such that |hz, y ∗ i| = ky ∗ kkzk. Let hz, y ∗ i = eiθ |hz, y ∗ i| . Therefore

−iθ  e z, y ∗ = |hz, y ∗ i| = ky ∗ kkzk = ky ∗ kke−iθ zk,   ∗ f (ky ∗ k) ∗ ∗ −iθ f (ky k) = z, y ky kke−iθ zk. e kzk1 kzk1

(2.7) (2.8)

From (2.7) we conclude that

−iθ



e z + λw ≥ e−iθ z

(2.9)

for w ∈ ker(y ∗ ) and λ ∈ C. Equation (2.9) implies that



−iθ f (ky ∗ k)

−iθ f (ky ∗ k)

e

≥ e

z + λw z

kzk1 kzk1

for w ∈ ker(y ∗ ) and λ ∈ C. Furthermore, w w  w w w −iθ f (ky ∗ k) w w −iθ f (ky ∗ k) w w w we = ϕ we z zw w kzk1 w∗ kzk1 w1

(2.10)

= ϕ (f (ky ∗ k)) = ky ∗ k.

From Theorem (2.5) we infer that J is surjective.



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Acknowledgements The authors were supported by the Polish Ministry of Sciences and Higher Education. References [1] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396–414. [2] S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers, Hauppauge, New York, 2004. [3] G. D. Faulkner, Representation of linear functionals in a Banach space, Rocky Mountain J. Math., 7 (1977), 789–792. [4] J. R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129 (1961), 436–446. [5] R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), 265–292. [6] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I, Bull. Amer. Math. Soc. (N.S.), 1 (1979), 230–232. [7] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), 29–43. [8] A. Misiak, n-inner product spaces, Math. Nachr., 140 (1989), 299–319. [9] V. Smulian: Sur la d´ erivabilit´ e de la norme dans l’espace de Banach, Dokl. Akad. Nauk SSSR, 27 (1940), 643–648. [10] E. Torrance, Strictly convex spaces via semi-inner-product space orthogonality, Proc. Amer. Math. Soc., 26 (1970), 108–110. [11] H. Zhang and J. Zhang, Generalized Semi-inner Products and Applications to Regularized Learning, J. Math. Analysis and Appl., 372 (2010), 181–196.

EWA SZLACHTOWSKA, AGH University of Science and Technology, Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Krak´ow, POLAND. [email protected]. DOMINIK MIELCZAREK, AGH University of Science and Technology, Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Krak´ow, POLAND. [email protected].