# A Reverse Theorem on the·-w* Continuity of the Dual Map

Authors: Mienie de Kock; Francisco Javier García-Pacheco
Journal: Journal of Function Spaces
 (2015)

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## Abstract

Given a Banach spaceX, x∈𝖲X, and 𝖩Xx=x*∈𝖲X*:x*x=1, we define the set 𝖩X*x of all x*∈𝖲X* for which there exist two sequences xnn∈N⊆𝖲X∖{x} and xn*n∈N⊆𝖲X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. We prove that if X is separable and reflexive and  X* enjoys the Radon-Riesz property, then 𝖩X*x is contained in the boundary of 𝖩Xx relative to 𝖲X*. We also show that if X is infinite dimensional and separable, then there exists an equivalent norm on X such that the interior of 𝖩Xx relative to 𝖲X* is contained in 𝖩X*x.


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## Body

## 1. Preliminaries and Background

Recall that a pointx in the unit sphere SX of a real or complex normed space X 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 JXx. The set of smooth points of the (closed) unit ball BX of X is usually denoted as smoBX. X is said to be smooth provided that SX=smoBX. If X is smooth, then the dual map of X is defined as the map JX:X→X* such that JXx=x and JXxx=x2 for all x∈X. It is well known that the dual map is ·-w* continuous and that JXλx=λ¯JXx for all λ∈C and all x∈X. We refer the reader to [1, 2] for a better perspective on these concepts.On the other hand, recall that a normed space is said to be rotund (or strictly convex) provided that its unit sphere is free of nontrivial segments. It is well known among Banach Space Geometers that smoothness and rotundity are dual concepts in the following sense: if a dual space is rotund (smooth), then the predual is smooth (rotund). The converse does not hold though. Next, we will gather some of the most relevant results in 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]).
LetX be a real or complex normed space. Then one has the following. (i)
IfX is separable, then X admits an equivalent norm so that both X and X* are rotund.(ii)
IfX is reflexive, then X admits an equivalent norm so that X is rotund and smooth.In casex∈SX is not a smooth point then JXx is defined as x-11∩BX*, that is, the set x*∈BX*:x*x=1. We will now continue with a brief introduction on faces and the impact of surjective linear isometries on them. The following definition is very well known amid Banach Space Geometers.Definition 2.
LetX be a real or complex normed space and consider a nonempty convex subset C of BX. Then one has the following. (i)
C is said to be a face of BX provided that C verifies the extremal condition with respect to BX; that is, if x,y∈BX and t∈0,1 with tx+1-ty∈C, then x,y∈C.(ii)
C is said to be an exposed face of BX provided that there exists f∈SX* such that C=Cf, where Cf≔f-11∩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 thatJXx=Cx for every x∈SX; that is, JXx is an exposed face of BX*. We will refer to these exposed faces as w*-exposed faces.A face, an exposed face, or aw*-exposed face, which is a singleton, is called an extreme point, an exposed point, or a w*-exposed point, respectively.Remark 3.
Observe the following, since we will use it in upcoming sections (see [1, 4]).(1)
A pointx∈smoBX if and only if Cx is a singleton, and, in this situation, Cx=JXx. So with this notation if x is a smooth point, then JXx is a w*-exposed point.(2)
Also note thatintSXC⊆smoBX for every proper face C.(3)
Assume now thatT:X→Y is a surjective linear isometry between the real or complex normed spaces X and Y. It is not difficult to check that if f∈SX*, then(1)TCf=CT-1*f.As mentioned before, in a smooth space ifxnn∈N converges to x, then the sequence JXxnn∈N is w*-convergent to JXx (see [1]). The main result in this paper is the converse to the previous statement (see Theorem 12).Theorem 4.
LetX be a separable real Banach space with dim⁡X>1. Consider x∈SX. Then one has the following. (1)
lbdSX*JXx⊆JX*x.(2)
IfX is reflexive and X* has the Radon-Riesz property, then JX*x⊆bdSX*JXx.(3)
IfX is infinite dimensional, then there exists an equivalent norm on X such that intSX*JXx⊆JX*x.More precisely, for a given real or complex Banach spaceX and x∈SX we will define the set JX*x of all x*∈BX* for which there exist two sequences xnn∈N⊆SX∖x and xn*n∈N⊆SX* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. As expected, bdSX*JXx and intSX*JXx denote the boundary and the interior, respectively, of JXx relative to SX*. The set lbdSX*JXx denotes the localized boundary of JXx (see Definition 9), that is, the set of all x*∈JXx for which there exists a dense subset D of X∖Rx with cardD=denX and satisfying that x*Yd∈bdSYd*JYdx for all d∈D, where Yd≔span⁡x,d (recall that denX stands for the density character of X).To conclude, we draw the reader’s attention to the fact that ifX is real and dim⁡X=1, then JX*x=⌀ for all x∈SX.


## 2. The Geometric Tools

Note that no complex Banach space admits proper faces with nonempty interior relative to the unit sphere in its unit ball (see [5, Theorem 2.1]). This is the reason for considering real Banach 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*-exposed faces 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)
JXx∩intSX*JXy=⌀.(2)
JXx∩JXy≠⌀ if and only if x,y⊂SX.(3)
Ifz∈SX∩x,y, then JXx∩JXy=JXz.Proof.
(
1
) Assume that there exists z*∈JXx∩intSX*JXy. As mentioned in the introduction, we have that intSX*JXy⊆smoBX*; therefore x=JX*z=y which contradicts the hypothesis of the lemma.
(
2
) If y*∈JXx∩JXy, then y*tx+1-ty=1 and tx+1-ty=1 for all t∈0,1. Conversely, if x,y⊂SX, then the Hahn-Banach Separation Theorem assures the existence of y*∈SX* such that (2)sup⁡y*UX≤inf⁡y*x,y,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 proper face relative to the unit sphere. Therefore,(3)extBX*∩JXx=extJXx⊆bdSX*JXx.Every w*-exposed face is w*-closed and thus w*-compact; therefore the Krein-Milman Theorem allows the existence of extreme points in every w*-exposed face (even more, every w*-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* continuous in smooth spaces.Lemma 7.
LetX be a real Banach space with dim⁡X>1 and fix an arbitrary element x∈SX. Then ⌀≠JX*x⊆JXx. As a consequence, if x∈smoBX, then JX*x=JXx.Proof.
Letx*∈JX*x. We can find two sequences xnn∈N⊆SX∖x and xn*n∈N⊆SX* such that xnn∈N converges to x, xn*n∈N has a subnet xni*i∈Iw*-convergent to x*, and xn*xn=1 for all n∈N. Next observe that for all i∈I we have (4)1-x*x=xni*xni-x*x≤xni*xni-xni*x+xni*x-x*x≤xni-x+xni*x-x*x,which implies that x*x=1 and thus x*∈JXx. In order to show that JX*x≠⌀, we simply start off with two sequences xnn∈N⊆SX∖x and xn*n∈N⊆SX* such that xnn∈N converges to x and xn*xn=1 for all n∈N. The w*-compactness of BX* assures that xn*n∈N has a subnet xni*i∈Iw*-convergent to x*. By definition x*∈JX*x.In general, no convexity properties are verified byJX*x even in finite dimensions. Indeed, if X≔l∞2 and x≔1,1, then JX*x=1,0,0,1.We will finish this section with a characterization of a smooth point in terms of localizing (linearly and topologically).Proposition 8.
LetX be a real Banach space. For a given point x∈SX, the following are equivalent. (1)
x∈smoBX.(2)
For everyd∈X, x is a smooth point of the unit ball of span⁡x,d.(3)
For every dense subsetD of X, x is a smooth point of the unit ball of span⁡x,d for all d∈D.(4)
There exists a dense subsetD of X with cardD=denX such that x is a smooth point of the unit ball of span⁡x,d for all d∈D.Proof.
(1)⇒(2) Letx*,y*∈Jspan⁡x,dx. By the Hahn-Banach Extension Theorem we may assume that x*,y*∈JXx, which implies that x*=y*.
(2)⇒(3) Immediate.
(3)⇒(4) Immediate.
(4)⇒(1) Letx*,y*∈JXx. For every d∈D we have that(5)x*span⁡x,d,y*span⁡x,d∈Jspan⁡x,dx,thus by hypothesis x*span⁡x,d=y*span⁡x,d for all d∈D. As a consequence, x*|D=y*|D and 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 Hausdorff locally compact topological space with more than one point, then the constant function1 is a uniformly nonsmooth point of BCK (recall that x∈SX is said to be a uniformly nonsmooth point provided that x is not a smooth point of the unit ball of span⁡x,y for all y∈SX∖±x).We refer the reader to [7, page 168] where a precise description of the smooth points of BCK is given, for K is a Hausdorff locally compact topological space.


## 3. The Main Result

We first define a new boundary for thew*-exposed faces other than the regular topological boundary.Definition 9.
LetX be a real Banach space and consider x∈SX. We define the localized boundary of JXx as the set lbdSX*JXx of all x*∈JXx for which there exists a dense subset D of X∖Rx with cardD=denX and satisfying that x*|Yd∈bdSYd*JYdx for all d∈D, where Yd≔span⁡x,d.Note that the localized boundary of aw*-exposed face is always contained in the topological boundary of that face relative to the dual unit sphere.Proposition 10.
LetX be a 2-dimensional real Banach space. For every x∈SX we have that lbdSX*JXx=bdSX*JXx⊆JX*x.Proof.
The2-dimensionality of X assures that (6)lbdSX*JXx=bdSX*JXx=x1*,x2*for some x1*≠x2*∈SX*. It is clear that we assume that x is not a smooth point in virtue of Lemma 7. In accordance to Mazur’s Theorem (see [8]), the separability of X allows for the density of the smooth points in the unit sphere; therefore (7)smoBX∩SX∩x1*-x2*-10,+∞isdenseinSX∩x1*-x2*-10,+∞,smoBX∩SX∩x1*-x2*-1-∞,0isdenseinSX∩x1*-x2*-1-∞,0.We can then find two sequences (8)xnn∈N⊂smoBX∩SX∩x1*-x2*-10,+∞,ynn∈N⊂smoBX∩SX∩x1*-x2*-1-∞,0both converging to x. Because of the compactness of SX* we can assume without loss of generality that the two sequences JXxnn∈N and JXynn∈N are, respectively, convergent to some a*≠b*∈x1*,x2* (recall the proof of Lemma 7).Remark 11.
In Proposition10 we have proved something more. Indeed, under the hypotheses of Proposition 10, if x*∈bdSX*JXx, then a sequence xnn∈N⊂smoBX∖x exists such that JXxnn∈Nw*-converges to x*.Prior to stating and proving the main result, we recall the following fact: a real or complex Banach spaceX is said to have the Radon-Riesz property provided that the following condition holds: if xnn∈N is w-convergent to x and xnn∈N is convergent to x, then xnn∈N converges to x.Theorem 12.
LetX be a separable real Banach space with dim⁡X>1. Consider x∈SX. Then one has the following. (1)
lbdSX*JXx⊆JX*x.(2)
IfX is reflexive and X* has the Radon-Riesz property, then JX*x⊆bdSX*JXx.(3)
IfX is infinite dimensional, then there exists an equivalent norm on X such that intSX*JXx⊆JX*x.Proof.
(
1
) Let x* be an element in lbdSX*JXx. By hypothesis there exists a dense sequence dnn∈N⊂X∖Rx such that x*|Yn∈bdSYn*JYnx for all n∈N, where Yn≔span⁡x,dn. By Proposition 10 for every n∈N, we have that x*|Yn∈JYn*x, so we can find xn∈SYn∖x and xn*∈SYn* such that xn*xn=1, x-xn<1/n and x*dn-xn*dn<1/n. By the Hahn-Banach Extension Theorem we may assume that xn*∈SX* for all n∈N. It only remains to show that xn*n∈N has a subnet w*-convergent to x*. In fact, we will show more; we will prove that the whole sequence xn*n∈N is w*-convergent to x*. Fix an arbitray y∈X. Notice that for every n∈N we have that (9)x*y-xn*y=x*y-x*dn+x*dn-xn*dn+xn*dn-xn*y<2y-dn+1n.At this stage, it is easy to understand that xn*ynn∈N converges to x*y.
(
2
) Assume to the contrary that there exists x*∈JX*x which is also in the interior of JXx relative to SX*. Notice that in this case x∉smoBX since the interior of JXx relative to SX* is not empty (which implies that JXx cannot be a singleton). Now consider two sequences xnn∈N⊆SX and xn*n∈N⊆SX* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. Observe that x*∈xn*:n∈N¯w*, so since the w* topology is metrizable in BX* (due to the fact that X is separable), there exists a subsequence xnk*k∈N of xn*n∈N which is w*-convergent to x*. Since X is reflexive, we have that the w* and the w topologies coincide on X*; therefore xnk*k∈N is w-convergent to x*. Since X* has the Radon-Riesz property, xnk*k∈N⊆SX*, and x*∈SX*, we deduce that xnk*k∈N is convergent to x*. This means that there exists m∈N such that xnk*k≥m⊂intSX*JXx. 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⊕1kerx* and X*=Rx*⊕∞kerx, so we will assume from now on that X is already endowed in such a way. Since X is infinite dimensional we have that SX* is w*-dense in BX* and so is NAX∩SX* since X is complete, where NAX denotes the set of norm-attaining functionals on X. We refer the reader to [9, Theorem 3.11] in order to take into consideration the fact that (10)intSX*JXx=x*+m*:m*∈Ukerx.Therefore, fix m*∈kerx which is also in the open unit ball of X* and denote y*≔x*+m*. We can find a sequence mn*n∈N⊆NAX∩Skerx which is w*-convergent to m* (remember that the w*-topology is metrizable in BX* due to the separability of X). For every n∈N there exists mn∈Skerx* such that mn*mn=1. Finally, consider for every n∈N the elements xn≔1-2-nx+2-nmn and xn*≔x*+mn*. By construction we have that xnn∈N⊆SX is convergent to x and xn*n∈N⊆SX* is w*-convergent to y*. As a consequence, y*∈JX*x.Note that if a (nonnecessarily convex) subsetE of SX verifies the extremal condition (see Definition 2) and E∩intSXC≠⌀ for some proper face C of BX, then C⊆E.Corollary 13.
LetX be an infinite dimensional separable real Banach space. For every x∈SX there exists an equivalent norm on X satisfying that JX*x does not verify the extremal condition.By combining Proposition10 and (2) of Theorem 12 we obtain the following corollary.Corollary 14.
LetX be a 2-dimensional real Banach space. For every x∈SX, JX*x=bdSX*JXx.



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