Research Article
A Reverse Theorem on the ‖⋅‖-
∗
Continuity of the Dual Map
Mienie de Kock
1
and Francisco Javier García-Pacheco
2
1
Department of Mathematics and Physics, Texas A&M University Central Texas, Killeen, TX 76548, USA
2
Department of Mathematics, University of Cadiz, 11519 Puerto Real, Spain
Correspondence should be addressed to Francisco Javier Garc´ ıa-Pacheco; garcia.pacheco@uca.es
Received 9 October 2014; Accepted 22 February 2015
Academic Editor: Henryk Hudzik
Copyright © 2015 M. de Kock and F. J. Garc´ ıa-Pacheco. his is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Given a Banach space , ∈ S
, and J
() = {
∗
∈ S
∗ :
∗
() = 1}, we deine the set J
∗
() of all
∗
∈ S
∗ for which there exist
two sequences (
)
∈N
⊆ S
\ {} and (
∗
)
∈N
⊆ S
∗ such that (
)
∈N
converges to , (
∗
)
∈N
has a subnet
∗
-convergent to
∗
,
and
∗
(
)=1 for all ∈ N. We prove that if is separable and relexive and
∗
enjoys the Radon-Riesz property, then J
∗
() is
contained in the boundary of J
() relative to S
∗ . We also show that if is ininite dimensional and separable, then there exists
an equivalent norm on such that the interior of J
() relative to S
∗ is contained in J
∗
().
1. Preliminaries and Background
Recall that a point in the unit sphere S
of a real or complex
normed space is said to be a smooth point of B
, provided
that there is only one functional in S
∗ attaining its norm at
. his unique functional is usually denoted by J
(). he set
of smooth points of the (closed) unit ball B
of is usually
denoted as smo(B
). is said to be smooth provided that
S
= smo(B
). If is smooth, then the dual map of is
deined as the map J
:→
∗
such that ‖J
()‖ = ‖‖
and J
()() = ‖‖
2
for all ∈. It is well known that the
dual map is ‖⋅‖-
∗
continuous and that J
() = J
() for
all ∈ C and all ∈. 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). he 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, heorem 1 (VII.4)] and [1, Corollary 4.3]).
heorem 1 (see [1, 3]). Let be a real or complex normed
space. hen one has the following.
(i) If is separable, then admits an equivalent norm so
that both and
∗
are rotund.
(ii) If is relexive, then admits an equivalent norm so
that is rotund and smooth.
In case ∈ S
is not a smooth point then J
() is deined
as
−1
(1)∩ B
∗ , that is, the set {
∗
∈ B
∗ :
∗
() = 1}. We will
now continue with a brief introduction on faces and the
impact of surjective linear isometries on them. he following
deinition is very well known amid Banach Space Geometers.
Deinition 2. Let be a real or complex normed space and
consider a nonempty convex subset of B
. hen one has the
following.
(i) is said to be a face of B
provided that veriies
the extremal condition with respect to B
; that is, if
, ∈ B
and ∈ (0, 1) with + (1 − ) ∈ , then
,∈.
(ii) is said to be an exposed face of B
provided that
there exists ∈ S
∗ such that = C
, where C
:=
−1
(1) ∩ B
.
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
() = C
for every ∈ S
; that is, J
() is
Hindawi Publishing Corporation
Journal of Function Spaces
Volume 2015, Article ID 864173, 4 pages
http://dx.doi.org/10.1155/2015/864173