On Norm Attaining Functionals
Pepe Moreno
ABSTRACT.
Consider a Banach space X and a bounded linear functional
f . The norm of f is defined as the supremum of
f over the unit ball of X . When this supremum is
attained, we say that f attains its norm. We denote by
NA the set of norm attaining functionals. The classical James' Theorem
asserts that X is reflexive if and only if every functionals
attains its norm. On the other hand, the Bishop-Phelps Theorem tell us
that NA is always dense in the dual space. It is our purpose to discuss
some topological properties of NA implying reflexivity. More precisely,
we are motivated by the following question: is a Banach space reflexive
provided the set NA has nonempty interior? It can be easily seen that
the answer is negative when the topology is the norm topology. This
situation suggests, at least, two possibilities. First, to investigate
which additional geometric conditions imply a positive answer. Second,
to consider the same problem for the weak or the weak* topologies.
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