research article a reverse theorem on the continuity of the dual...

Research ArticleA Reverse Theorem on the ‖ ⋅ ‖-𝑤∗ Continuity of the Dual Map

Mienie de Kock1 and Francisco Javier García-Pacheco2

1Department of Mathematics and Physics, Texas A&M University Central Texas, Killeen, TX 76548, USA2Department of Mathematics, University of Cadiz, 11519 Puerto Real, Spain

Correspondence should be addressed to Francisco Javier Garćıa-Pacheco; [email protected]

Received 9 October 2014; Accepted 22 February 2015

Academic Editor: Henryk Hudzik

Copyright © 2015 M. de Kock and F. J. Garćıa-Pacheco. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Given a Banach space 𝑋, 𝑥 ∈ S𝑋, and J

𝑋(𝑥) = {𝑥

∗∈ S𝑋∗ : 𝑥∗(𝑥) = 1}, we define the set J∗

𝑋(𝑥) of all 𝑥∗ ∈ S

𝑋∗ for which there exist

two sequences (𝑥𝑛)𝑛∈N ⊆ S𝑋 \ {𝑥} and (𝑥

∗

𝑛)𝑛∈N ⊆ S𝑋∗ such that (𝑥𝑛)𝑛∈N converges to 𝑥, (𝑥

∗

𝑛)𝑛∈N has a subnet 𝑤

∗-convergent to 𝑥∗,and 𝑥∗

𝑛(𝑥𝑛) = 1 for all 𝑛 ∈ N. We prove that if 𝑋 is separable and reflexive and 𝑋∗ enjoys the Radon-Riesz property, then J∗

𝑋(𝑥) is

contained in the boundary of J𝑋(𝑥) relative to S

𝑋∗ . We also show that if 𝑋 is infinite dimensional and separable, then there exists

an equivalent norm on 𝑋 such that the interior of J𝑋(𝑥) relative to S

𝑋∗ is contained in J∗

𝑋(𝑥).

1. Preliminaries and Background

Recall that a point 𝑥 in the unit sphere S𝑋of a real or complex

normed space𝑋 is said to be a smooth point of B𝑋, provided

that there is only one functional in S𝑋∗ attaining its norm at

𝑥. This unique functional is usually denoted by J𝑋(𝑥). The set

of smooth points of the (closed) unit ball B𝑋of 𝑋 is usually

denoted as smo(B𝑋). 𝑋 is said to be smooth provided that

S𝑋

= smo(B𝑋). If 𝑋 is smooth, then the dual map of 𝑋 is

defined as the map J𝑋

: 𝑋 → 𝑋∗ such that ‖J

𝑋(𝑥)‖ = ‖𝑥‖

and J𝑋(𝑥)(𝑥) = ‖𝑥‖

2 for all 𝑥 ∈ 𝑋. It is well known that thedual map is ‖ ⋅ ‖-𝑤∗ continuous and that J

𝑋(𝜆𝑥) = 𝜆J

𝑋(𝑥) for

all𝜆 ∈ C and all𝑥 ∈ 𝑋.We refer the reader to [1, 2] for a betterperspective on these concepts.

On the other hand, recall that a normed space is said to berotund (or strictly convex) provided that its unit sphere is freeof nontrivial segments. It is well known among Banach SpaceGeometers that smoothness and rotundity are dual conceptsin the following sense: if a dual space is rotund (smooth), thenthe predual is smooth (rotund). The converse does not holdthough. Next, we will gather some of themost relevant resultsin terms of rotund and smooth renormings into the following(see [3, Theorem 1 (VII.4)] and [1, Corollary 4.3]).

Theorem 1 (see [1, 3]). Let 𝑋 be a real or complex normedspace. Then one has the following.

(i) If𝑋 is separable, then𝑋 admits an equivalent norm sothat both 𝑋 and 𝑋∗ are rotund.

(ii) If 𝑋 is reflexive, then 𝑋 admits an equivalent norm sothat 𝑋 is rotund and smooth.

In case 𝑥 ∈ S𝑋is not a smooth point then J

𝑋(𝑥) is defined

as 𝑥−1(1)∩B𝑋∗ , that is, the set {𝑥∗ ∈ B

𝑋∗ : 𝑥∗(𝑥) = 1}.Wewill

now continue with a brief introduction on faces and theimpact of surjective linear isometries on them.The followingdefinition is very well known amid Banach Space Geometers.

Definition 2. Let 𝑋 be a real or complex normed space andconsider a nonempty convex subset𝐶 ofB

𝑋.Then one has the

following.

(i) 𝐶 is said to be a face of B𝑋provided that 𝐶 verifies

the extremal condition with respect to B𝑋; that is, if

𝑥, 𝑦 ∈ B𝑋and 𝑡 ∈ (0, 1) with 𝑡𝑥 + (1 − 𝑡)𝑦 ∈ 𝐶, then

𝑥, 𝑦 ∈ 𝐶.(ii) 𝐶 is said to be an exposed face of B

𝑋provided that

there exists 𝑓 ∈ S𝑋∗ such that 𝐶 = C

𝑓, where C

𝑓:=

𝑓−1

(1) ∩ B𝑋.

It is immediate that every exposed face is a proper face,and every proper face must be contained in the unit sphere.Also notice that J

𝑋(𝑥) = C

𝑥for every 𝑥 ∈ S

𝑋; that is, J

𝑋(𝑥) is

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015, Article ID 864173, 4 pageshttp://dx.doi.org/10.1155/2015/864173

2 Journal of Function Spaces

an exposed face of B𝑋∗ . We will refer to these exposed faces

as 𝑤∗-exposed faces.A face, an exposed face, or a 𝑤∗-exposed face, which is a

singleton, is called an extreme point, an exposed point, or a𝑤∗-exposed point, respectively.

Remark 3. Observe the following, since we will use it inupcoming sections (see [1, 4]).

(1) A point 𝑥 ∈ smo(B𝑋) if and only if C

𝑥is a singleton,

and, in this situation, C𝑥

= {J𝑋(𝑥)}. So with this

notation if 𝑥 is a smooth point, then J𝑋(𝑥) is a 𝑤∗-

exposed point.(2) Also note that intS

𝑋

(𝐶) ⊆ smo(B𝑋) for every proper

face 𝐶.(3) Assume now that 𝑇 : 𝑋 → 𝑌 is a surjective linear

isometry between the real or complex normed spaces𝑋 and 𝑌. It is not difficult to check that if 𝑓 ∈ S

𝑋∗ ,

then

𝑇 (C𝑓) = C

(𝑇−1)∗(𝑓)

. (1)

As mentioned before, in a smooth space if (𝑥𝑛)𝑛∈N con-

verges to 𝑥, then the sequence (J𝑋(𝑥𝑛))𝑛∈N is 𝑤

∗-convergentto J𝑋(𝑥) (see [1]).Themain result in this paper is the converse

to the previous statement (see Theorem 12).

Theorem 4. Let 𝑋 be a separable real Banach space withdim(𝑋) > 1. Consider 𝑥 ∈ S

𝑋. Then one has the following.

(1) lbdS𝑋∗(J𝑋(𝑥)) ⊆ J∗

𝑋(𝑥).

(2) If 𝑋 is reflexive and 𝑋∗ has the Radon-Riesz property,then J∗

𝑋(𝑥) ⊆ bdS

𝑋∗(J𝑋(𝑥)).

(3) If 𝑋 is infinite dimensional, then there exists anequivalent norm on𝑋 such that intS

𝑋∗(J𝑋(𝑥)) ⊆ J∗

𝑋(𝑥).

More precisely, for a given real or complex Banach space𝑋 and 𝑥 ∈ S

𝑋we will define the set J∗

𝑋(𝑥) of all 𝑥∗ ∈ B

𝑋∗

for which there exist two sequences (𝑥𝑛)𝑛∈N ⊆ S𝑋 \ {𝑥} and

(𝑥∗

𝑛)𝑛∈N ⊆ S𝑋∗ such that (𝑥𝑛)𝑛∈N converges to 𝑥, (𝑥

∗

𝑛)𝑛∈N has

a subnet 𝑤∗-convergent to 𝑥∗, and 𝑥∗𝑛(𝑥𝑛) = 1 for all 𝑛 ∈

N. As expected, bdS𝑋∗(J𝑋(𝑥)) and intS

𝑋∗(J𝑋(𝑥)) denote the

boundary and the interior, respectively, of J𝑋(𝑥) relative to

S𝑋∗ . The set lbdS

𝑋∗(J𝑋(𝑥)) denotes the localized boundary of

J𝑋(𝑥) (see Definition 9), that is, the set of all 𝑥∗ ∈ J

𝑋(𝑥) for

which there exists a dense subset𝐷 of𝑋\R𝑥with card(𝐷) =den(𝑋) and satisfying that 𝑥∗|

𝑌𝑑

∈ bdS𝑌∗

𝑑

(J𝑌𝑑

(𝑥)) for all 𝑑 ∈𝐷, where 𝑌

𝑑:= span{𝑥, 𝑑} (recall that den(𝑋) stands for the

density character of 𝑋).To conclude, we draw the reader’s attention to the fact that

if 𝑋 is real and dim(𝑋) = 1, then J∗𝑋(𝑥) = ⌀ for all 𝑥 ∈ S

𝑋.

2. The Geometric Tools

Note that no complex Banach space admits proper faces withnonempty interior relative to the unit sphere in its unit ball(see [5, Theorem 2.1]). This is the reason for considering realBanach spaces only. On our journey to proving the main

result, we will utilize several technical lemmas and remarks.The first lemma involves the intersections of 𝑤∗-exposedfaces and the disjointness of their interiors.

Lemma 5. Let𝑋 be a real Banach space and consider 𝑥 ̸= 𝑦 ∈S𝑋. Then one has the following.

(1) J𝑋(𝑥) ∩ intS

𝑋∗(J𝑋(𝑦)) = ⌀.

(2) J𝑋(𝑥) ∩ J

𝑋(𝑦) ̸= ⌀ if and only if [𝑥, 𝑦] ⊂ S

𝑋.

(3) If 𝑧 ∈ S𝑋

∩ (𝑥, 𝑦), then J𝑋(𝑥) ∩ J

𝑋(𝑦) = J

𝑋(𝑧).

Proof. (1)Assume that there exists 𝑧∗ ∈ J𝑋(𝑥)∩intS

𝑋∗(J𝑋(𝑦)).

As mentioned in the introduction, we have that intS𝑋∗(J𝑋(𝑦))

⊆ smo(B𝑋∗); therefore 𝑥 = J

𝑋∗(𝑧) = 𝑦 which contradicts the

hypothesis of the lemma.(2) If 𝑦∗ ∈ J

𝑋(𝑥) ∩ J

𝑋(𝑦), then 𝑦∗(𝑡𝑥 + (1 − 𝑡)𝑦) = 1 and

‖𝑡𝑥+ (1− 𝑡)𝑦‖ = 1 for all 𝑡 ∈ [0, 1]. Conversely, if [𝑥, 𝑦] ⊂ S𝑋,

then the Hahn-Banach SeparationTheorem assures the exis-tence of 𝑦∗ ∈ S

𝑋∗ such that

sup𝑦∗ (U𝑋) ≤ inf 𝑦∗ ([𝑥, 𝑦]) , (2)

which immediately implies that 𝑦∗(𝑦) = 𝑦∗(𝑥) = 1.(3) The proof uses a similar argument as in the previous

item.

Remark 6. No extreme point can lie in the interior of a properface relative to the unit sphere. Therefore,

ext (B𝑋∗) ∩ J𝑋 (

𝑥) = ext (J𝑋 (𝑥)) ⊆ bdS𝑋∗(J𝑋 (

𝑥)) .

(3)

Every 𝑤∗-exposed face is 𝑤∗-closed and thus 𝑤∗-compact;therefore the Krein-MilmanTheorem allows the existence ofextreme points in every 𝑤∗-exposed face (even more, every𝑤∗-exposed face is the 𝑤∗-closed convex hull of its extreme

points).The second technical lemma is a slight generalization of

the well-known fact that the dual map is ‖ ⋅ ‖-𝑤∗ continuousin smooth spaces.

Lemma 7. Let𝑋 be a real Banach space with dim(𝑋) > 1 andfix an arbitrary element 𝑥 ∈ S

𝑋. Then ⌀ ̸= J∗

𝑋(𝑥) ⊆ J

𝑋(𝑥). As

a consequence, if 𝑥 ∈ smo(B𝑋), then J∗

𝑋(𝑥) = J

𝑋(𝑥).

Proof. Let 𝑥∗ ∈ J∗𝑋(𝑥). We can find two sequences (𝑥

𝑛)𝑛∈N ⊆

S𝑋

\ {𝑥} and (𝑥∗𝑛)𝑛∈N ⊆ S𝑋∗ such that (𝑥𝑛)𝑛∈N converges

to 𝑥, (𝑥∗𝑛)𝑛∈N has a subnet (𝑥

∗

𝑛𝑖

)𝑖∈𝐼

𝑤∗-convergent to 𝑥∗, and

𝑥∗

𝑛(𝑥𝑛) = 1 for all 𝑛 ∈ N. Next observe that for all 𝑖 ∈ 𝐼 we

have1 − 𝑥∗(𝑥)

=

𝑥∗

𝑛𝑖

(𝑥𝑛𝑖

) − 𝑥∗(𝑥)

≤

𝑥∗

𝑛𝑖

(𝑥𝑛𝑖

) − 𝑥∗

𝑛𝑖

(𝑥)

+

𝑥∗

𝑛𝑖

(𝑥) − 𝑥∗(𝑥)

≤

𝑥𝑛𝑖

− 𝑥

+

𝑥∗

𝑛𝑖

(𝑥) − 𝑥∗(𝑥)

,

(4)

which implies that 𝑥∗(𝑥) = 1 and thus 𝑥∗ ∈ J𝑋(𝑥). In order to

show that J∗𝑋(𝑥) ̸= ⌀, we simply start off with two sequences

(𝑥𝑛)𝑛∈N ⊆ S𝑋 \ {𝑥} and (𝑥

∗

𝑛)𝑛∈N ⊆ S𝑋∗ such that (𝑥𝑛)𝑛∈N

converges to 𝑥 and 𝑥∗𝑛(𝑥𝑛) = 1 for all 𝑛 ∈ N. The

Journal of Function Spaces 3

𝑤∗-compactness of B

𝑋∗ assures that (𝑥∗

𝑛)𝑛∈N has a subnet

(𝑥∗

𝑛𝑖

)𝑖∈𝐼

𝑤∗-convergent to 𝑥∗. By definition 𝑥∗ ∈ J∗

𝑋(𝑥).

In general, no convexity properties are verified by J∗𝑋(𝑥)

even in finite dimensions. Indeed, if 𝑋 := ℓ2∞

and 𝑥 := (1, 1),then J∗

𝑋(𝑥) = {(1, 0), (0, 1)}.

We will finish this section with a characterization of asmooth point in terms of localizing (linearly and topologi-cally).

Proposition 8. Let𝑋 be a real Banach space. For a given point𝑥 ∈ S𝑋, the following are equivalent.

(1) 𝑥 ∈ smo(B𝑋).

(2) For every 𝑑 ∈ 𝑋, 𝑥 is a smooth point of the unit ball ofspan{𝑥, 𝑑}.

(3) For every dense subset 𝐷 of 𝑋, 𝑥 is a smooth point ofthe unit ball of span{𝑥, 𝑑} for all 𝑑 ∈ 𝐷.

(4) There exists a dense subset 𝐷 of 𝑋 with 𝑐𝑎𝑟𝑑(𝐷) =den(𝑋) such that 𝑥 is a smooth point of the unit ballof span{𝑥, 𝑑} for all 𝑑 ∈ 𝐷.

Proof. (1)⇒(2) Let𝑥∗, 𝑦∗ ∈ Jspan{𝑥,𝑑}(𝑥). By theHahn-BanachExtension Theorem we may assume that 𝑥∗, 𝑦∗ ∈ J

𝑋(𝑥),

which implies that 𝑥∗ = 𝑦∗.(2)⇒(3) Immediate.(3)⇒(4) Immediate.(4)⇒(1) Let 𝑥∗, 𝑦∗ ∈ J

𝑋(𝑥). For every 𝑑 ∈ 𝐷 we have that

𝑥∗span{𝑥,𝑑} , 𝑦

∗span{𝑥,𝑑} ∈ Jspan{𝑥,𝑑} (𝑥) , (5)

thus by hypothesis 𝑥∗|span{𝑥,𝑑} = 𝑦∗|span{𝑥,𝑑} for all 𝑑 ∈ 𝐷. As a

consequence, 𝑥∗|𝐷

= 𝑦∗|𝐷and so 𝑥∗ = 𝑦∗ due to the density

of 𝐷 in 𝑋.

In [6, Remark 2.9(1)] it is shown that if 𝐾 is a Hausdorfflocally compact topological space with more than one point,then the constant function 1 is a uniformly nonsmooth pointof BC(𝐾) (recall that 𝑥 ∈ S𝑋 is said to be a uniformlynonsmooth point provided that 𝑥 is not a smooth point of theunit ball of span{𝑥, 𝑦} for all 𝑦 ∈ S

𝑋\ {±𝑥}).

We refer the reader to [7, page 168] where a precisedescription of the smooth points of BC(𝐾) is given, for 𝐾 isa Hausdorff locally compact topological space.

3. The Main Result

We first define a new boundary for the 𝑤∗-exposed facesother than the regular topological boundary.

Definition 9. Let 𝑋 be a real Banach space and consider 𝑥 ∈S𝑋. We define the localized boundary of J

𝑋(𝑥) as the set

lbdS𝑋∗(J𝑋(𝑥)) of all 𝑥∗ ∈ J

𝑋(𝑥) for which there exists a dense

subset𝐷 of𝑋\R𝑥with 𝑐𝑎𝑟𝑑(𝐷) = den(𝑋) and satisfying that𝑥∗|𝑌𝑑

∈ bdS𝑌∗

𝑑

(J𝑌𝑑

(𝑥)) for all 𝑑 ∈ 𝐷, where 𝑌𝑑:= span{𝑥, 𝑑}.

Note that the localized boundary of a 𝑤∗-exposed faceis always contained in the topological boundary of that facerelative to the dual unit sphere.

Proposition 10. Let 𝑋 be a 2-dimensional real Banachspace. For every 𝑥 ∈ S

𝑋we have that lbdS

𝑋∗(J𝑋(𝑥)) =

bdS𝑋∗(J𝑋(𝑥)) ⊆ J∗

𝑋(𝑥).

Proof. The 2-dimensionality of 𝑋 assures that

lbdS𝑋∗(J𝑋 (

𝑥)) = bdS𝑋∗(J𝑋 (

𝑥)) = {𝑥∗

1, 𝑥∗

2} (6)

for some 𝑥∗1

̸= 𝑥∗

2∈ S𝑋∗ . It is clear that we assume that 𝑥

is not a smooth point in virtue of Lemma 7. In accordance toMazur’sTheorem (see [8]), the separability of𝑋 allows for thedensity of the smooth points in the unit sphere; therefore

smo (B𝑋) ∩ S𝑋

∩ (𝑥∗

1− 𝑥∗

2)

−1

((0, +∞)) is dense in

S𝑋

∩ (𝑥∗

1− 𝑥∗

2)

−1

((0, +∞)) ,

smo (B𝑋) ∩ S𝑋

∩ (𝑥∗

1− 𝑥∗

2)

−1

((−∞, 0)) is dense in

S𝑋

∩ (𝑥∗

1− 𝑥∗

2)

−1

((−∞, 0)) .

(7)

We can then find two sequences

(𝑥𝑛)𝑛∈N

⊂ smo (B𝑋) ∩ S𝑋

∩ (𝑥∗

1− 𝑥∗

2)

−1

((0, +∞)) ,

(𝑦𝑛)𝑛∈N

⊂ smo (B𝑋) ∩ S𝑋

∩ (𝑥∗

1− 𝑥∗

2)

−1

((−∞, 0))

(8)

both converging to 𝑥. Because of the compactness of S𝑋∗ we

can assume without loss of generality that the two sequences(J𝑋(𝑥𝑛))𝑛∈N and (J𝑋(𝑦𝑛))𝑛∈N are, respectively, convergent to

some 𝑎∗ ̸= 𝑏∗ ∈ {𝑥∗1, 𝑥∗

2} (recall the proof of Lemma 7).

Remark 11. In Proposition 10 we have proved somethingmore. Indeed, under the hypotheses of Proposition 10, if 𝑥∗ ∈bdS𝑋∗(J𝑋(𝑥)), then a sequence (𝑥

𝑛)𝑛∈N ⊂ smo(B𝑋) \ {𝑥} exists

such that (J𝑋(𝑥𝑛))𝑛∈N 𝑤

∗-converges to 𝑥∗.

Prior to stating and proving the main result, we recall thefollowing fact: a real or complex Banach space 𝑋 is said tohave the Radon-Riesz property provided that the followingcondition holds: if (𝑥

𝑛)𝑛∈N is𝑤-convergent to 𝑥 and (‖𝑥𝑛‖)𝑛∈N

is convergent to ‖𝑥‖, then (𝑥𝑛)𝑛∈N converges to 𝑥.

Theorem 12. Let 𝑋 be a separable real Banach space withdim(𝑋) > 1. Consider 𝑥 ∈ S

𝑋. Then one has the following.

(1) lbdS𝑋∗(J𝑋(𝑥)) ⊆ J∗

𝑋(𝑥).

(2) If 𝑋 is reflexive and 𝑋∗ has the Radon-Riesz property,then J∗

𝑋(𝑥) ⊆ bdS

𝑋∗(J𝑋(𝑥)).

(3) If 𝑋 is infinite dimensional, then there exists anequivalent norm on𝑋 such that intS

𝑋∗(J𝑋(𝑥)) ⊆ J∗

𝑋(𝑥).

Proof. (1) Let 𝑥∗ be an element in lbdS𝑋∗(J𝑋(𝑥)). By hypoth-

esis there exists a dense sequence (𝑑𝑛)𝑛∈N ⊂ 𝑋 \R𝑥 such that

𝑥∗|𝑌𝑛

∈ bdS𝑌∗

𝑛

(J𝑌𝑛

(𝑥)) for all 𝑛 ∈ N, where 𝑌𝑛:= span{𝑥, 𝑑

𝑛}.

By Proposition 10 for every 𝑛 ∈ N, we have that 𝑥∗|𝑌𝑛

∈

J∗𝑌𝑛

(𝑥), so we can find 𝑥𝑛

∈ S𝑌𝑛

\ {𝑥} and 𝑥∗𝑛

∈ S𝑌∗

𝑛

such that𝑥∗

𝑛(𝑥𝑛) = 1, ‖𝑥 − 𝑥

𝑛‖ < 1/𝑛 and |𝑥∗(𝑑

𝑛) − 𝑥∗

𝑛(𝑑𝑛)| < 1/𝑛.

By the Hahn-Banach Extension Theorem we may assume

4 Journal of Function Spaces

that 𝑥∗𝑛

∈ S𝑋∗ for all 𝑛 ∈ N. It only remains to show that

(𝑥∗

𝑛)𝑛∈N has a subnet 𝑤

∗-convergent to 𝑥∗. In fact, we willshow more; we will prove that the whole sequence (𝑥∗

𝑛)𝑛∈N

is𝑤∗-convergent to 𝑥∗. Fix an arbitray 𝑦 ∈ 𝑋. Notice that forevery 𝑛 ∈ N we have that𝑥∗(𝑦) − 𝑥

∗

𝑛(𝑦)

=

𝑥∗(𝑦) − 𝑥

∗(𝑑𝑛)+

𝑥∗(𝑑𝑛) − 𝑥∗

𝑛(𝑑𝑛)

+𝑥∗

𝑛(𝑑𝑛) − 𝑥∗

𝑛(𝑦)

< 2

𝑦 − 𝑑𝑛

+

1

𝑛

.

(9)

At this stage, it is easy to understand that (𝑥∗𝑛(𝑦𝑛))𝑛∈N

converges to 𝑥∗(𝑦).(2) Assume to the contrary that there exists 𝑥∗ ∈ J∗

𝑋(𝑥)

which is also in the interior of J𝑋(𝑥) relative to S

𝑋∗ . Notice

that in this case 𝑥 ∉ smo(B𝑋) since the interior of J

𝑋(𝑥)

relative to S𝑋∗ is not empty (which implies that J

𝑋(𝑥) cannot

be a singleton). Now consider two sequences (𝑥𝑛)𝑛∈N ⊆ S𝑋

and (𝑥∗𝑛)𝑛∈N ⊆ S𝑋∗ such that (𝑥𝑛)𝑛∈N converges to 𝑥, (𝑥

∗

𝑛)𝑛∈N

has a subnet 𝑤∗-convergent to 𝑥∗, and 𝑥∗𝑛(𝑥𝑛) = 1 for all 𝑛 ∈

N. Observe that𝑥∗ ∈ {𝑥∗𝑛

: 𝑛 ∈ N}𝑤∗

, so since the𝑤∗ topologyis metrizable in B

𝑋∗ (due to the fact that 𝑋 is separable),

there exists a subsequence (𝑥∗𝑛𝑘

)𝑘∈N of (𝑥

∗

𝑛)𝑛∈N which is 𝑤

∗-convergent to 𝑥∗. Since 𝑋 is reflexive, we have that the 𝑤∗and the𝑤 topologies coincide on𝑋∗; therefore (𝑥∗

𝑛𝑘

)𝑘∈N is𝑤-

convergent to 𝑥∗. Since 𝑋∗ has the Radon-Riesz property,(𝑥∗

𝑛𝑘

)𝑘∈N ⊆ S𝑋∗ , and 𝑥

∗∈ S𝑋∗ , we deduce that (𝑥∗

𝑛𝑘

)𝑘∈N is

convergent to 𝑥∗. This means that there exists 𝑚 ∈ N suchthat (𝑥∗

𝑛𝑘

)𝑘≥𝑚

⊂ intS𝑋∗(J𝑋(𝑥)).This is impossible by Lemma 5.

(3) Fix an arbitrary 𝑥∗ ∈ S𝑋∗ such that 𝑥∗(𝑥) = 1.

Observe that 𝑋 can be equivalently renormed so that 𝑋 =R𝑥⊕1ker(𝑥∗) and 𝑋∗ = R𝑥∗⊕

∞ker(𝑥), so we will assume

fromnowon that𝑋 is already endowed in such away. Since𝑋is infinite dimensional we have that S

𝑋∗ is 𝑤∗-dense in B

𝑋∗

and so is NA(𝑋) ∩ S𝑋∗ since 𝑋 is complete, where NA(𝑋)

denotes the set of norm-attaining functionals on 𝑋. Werefer the reader to [9, Theorem 3.11] in order to take intoconsideration the fact that

intS𝑋∗(J𝑋 (

𝑥)) = {𝑥∗+ 𝑚∗

: 𝑚∗

∈ Uker(𝑥)} . (10)

Therefore, fix 𝑚∗ ∈ ker(𝑥) which is also in the open unit ballof 𝑋∗ and denote 𝑦∗ := 𝑥∗ + 𝑚∗. We can find a sequence(𝑚∗

𝑛)𝑛∈N ⊆ NA(𝑋) ∩ Sker(𝑥) which is 𝑤

∗-convergent to 𝑚∗(remember that the 𝑤∗-topology is metrizable in B

𝑋∗ due to

the separability of 𝑋). For every 𝑛 ∈ N there exists 𝑚𝑛

∈

Sker(𝑥∗) such that 𝑚∗

𝑛(𝑚𝑛) = 1. Finally, consider for every

𝑛 ∈ N the elements𝑥𝑛:= (1−2

−𝑛)𝑥+2−𝑛

𝑚𝑛and𝑥∗𝑛

:= 𝑥∗+𝑚∗

𝑛.

By construction we have that (𝑥𝑛)𝑛∈N ⊆ S𝑋 is convergent to 𝑥

and (𝑥∗𝑛)𝑛∈N ⊆ S𝑋∗ is𝑤

∗-convergent to 𝑦∗. As a consequence,𝑦∗

∈ J∗𝑋(𝑥).

Note that if a (nonnecessarily convex) subset 𝐸 of S𝑋

verifies the extremal condition (see Definition 2) and 𝐸 ∩intS𝑋

(𝐶) ̸= ⌀ for some proper face 𝐶 of B𝑋, then 𝐶 ⊆ 𝐸.

Corollary 13. Let 𝑋 be an infinite dimensional separable realBanach space. For every 𝑥 ∈ S

𝑋there exists an equivalent

norm on 𝑋 satisfying that J∗𝑋(𝑥) does not verify the extremal

condition.

By combining Proposition 10 and (2) of Theorem 12 weobtain the following corollary.

Corollary 14. Let𝑋 be a 2-dimensional real Banach space. Forevery 𝑥 ∈ S

𝑋, J∗𝑋(𝑥) = bdS

𝑋∗

(J𝑋(𝑥)).

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

[1] R. Deville, G. Godefroy, and V. Zizler, Smoothness and Renorm-ings in Banach Spaces, vol. 64 of Pitman Monographs andSurveys in Pure and Applied Mathematics, Longman, Harlow,UK, 1993.

[2] R. E. Megginson, An Introduction to Banach Space Theory, vol.183 of Graduate Texts in Mathematics, Springer, New York, NY,USA, 1998.

[3] M. M. Day, Normed Linear Spaces, vol. 21 of Ergebnisse derMathematik und ihrer Grenzgebiete, Springer, New York, NY,USA, 3rd edition, 1973.

[4] J. Diestel, Geometry of Banach Spaces—Selected Topics, vol. 485of Lecture Notes in Mathematics, Springer, Berlin, Germany,1975.

[5] F. J. Garćıa-Pacheco and A. Miralles, “Real renormings oncomplex Banach spaces,” Chinese Annals of Mathematics. SeriesB, vol. 29, no. 3, pp. 239–246, 2008.

[6] F. J. Garćıa-Pacheco, “A short note about exposed points in realBanach spaces,” Acta Mathematica Scientia B, vol. 28, no. 4, pp.797–800, 2008.

[7] S. Banach,Théorie des Opérations Linéaires, Chelsea Publishing,New York, NY, USA, 1978.

[8] S. Mazur, “Über konvexe mengen in linearem normieten Rau-men,” Studia Mathematica, vol. 4, pp. 70–84, 1933.

[9] M. D. Acosta, A. Aizpuru, R. M. Aron, and F. J. Garćıa-Pacheco,“Functionals that do not attain their norm,” Bulletin of theBelgian Mathematical Society—Simon Stevin, vol. 14, no. 3, pp.407–418, 2007.

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