Positivity 8: 283–296, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Weak-Polynomial Convergence on Spaces
and JOSÉ G. LLAVONA
Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires,
(1428) Buenos Aires, Argentina. E-mail: email@example.com
Departamento de Análisis
Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid,
España. E-mail: JL_Llavona@mat.ucm.es
(Received 9 December 2002; accepted 23 March 2003)
Abstract. This paper is concerned with the study of the set P
0, when P varies over all ortho-
gonally additive polynomials on
spaces. We apply our results to obtain characterizations
of the weak-polynomial topologies associated to this class of polynomials.
Key words: polynomials on Banach spaces, weak-polynomial topologies, zeros of polynomials on
Before going into the problem, we discuss some preliminaries and ﬁx notation.
Throughout, X will be a real Banach space and X
its dual. We are going to consider
only continuous polynomials and, as usual, denote by P
X the space of all n-
homogeneous continuous scalar-valued polynomials with domain X PX will be
the space of all continuous scalar-valued polynomials deﬁned on X. P
X is a
Banach space endowed with the norm P=supPx x 1
We may deﬁne various topologies on X in terms of convergence of nets: the
strong topology in which a net x
→x if and only if x
−x→0, the weak
(w) topology where x
→x if and only if x
−x→0 for all ∈ X
weak polynomial (wp) topology (see Carne et al., 1989) with convergence given by
→x if and only if Px
−x→0 for all P∈PX or equivalently for all
X for all n∈ The weak polynomial topology was studied in Aron et
al. (1991); Biström et al. (1998); Davie and Gamelin (1989); González et al. (1997)
and Gutiérrez and Llavona (1997).
It is easy to see that x
→x. For any inﬁnite
dimensional complex Hilbert space H the wp-topology is not linear even when
restricted to the unit ball of H (see, Aron et al., 1991). New examples of both real
and complex Banach spaces X such that the wp-topology is not linear are given in
Biström et al. (1998) and Castillo et al. (1999).