Positivity 8: 297–304, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Polynomials, Symmetric Multilinear Forms
and Weak Compactness
MANUEL RUIZ GALÁN
Departamento de Matemática Aplicada, E.U. Arquitectura Técnica, Universidad de Granada, c/
Severo Ochoa s/n, 18071 Granada, Spain. E-mail: firstname.lastname@example.org
(Received 26 March 2002; accepted 5 January 2003)
Abstract. In this paper we obtain some versions of weak compactness James’ theorem, replacing
bounded linear functionals by polynomials and symmetric multilinear forms.
Mathematics Subject Classiﬁcation (1991): 46B10, 46B50, 46G25
Key words: norm attaining polynomials and symmetric multilinear forms on a Banach space, weak
James’ theorem  characterizes the reﬂexivity of a Banach space by means of
norm attaining functionals. To be more precise, if we write E
for the dual space
of a Banach space E, then a functional x
is said to attain its norm provided
that there exists x
(closed unit ball of E) such that
James’ theorem asserts — in its most important special case — that a Banach space
is reﬂexive if (and only if) each functional on it attains the norm. The aim of this
work is to give some James type results, considering polynomials and symmetric
We shall restrict ourselves, for the sake of simplicity, to the study of real Banach
spaces, although the proof of the results can be easily adapted to the complex case.
We shall use the notation “
co ” and “co
” to stand “(norm) closed convex hull”
and “weak-∗ closed convex hull,” respectively.
2 A version of James’ theorem for polynomials
In this section, we discuss, in particular, the reﬂexivity of a Banach space using
continuous homogeneous polynomials which attain their norms. A related ques-