PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, January 1999, Pages 241–250 S 0002-9939(99)04485-8 A HAHN-BANACH THEOREM FOR INTEGRAL POLYNOMIALS DANIEL CARANDO AND IGNACIO ZALDUENDO (Communicated by Theodore W. Gamelin) Abstract. We study the problem of extendibility of polynomials over Banach spaces: when can a polynomial defined over a Banach space be extended to a polynomial over any larger Banach space? To this end, we identify all spaces of polynomials as the topological duals of a space S spanned by evaluations, with Hausdorff locally convex topologies. We prove that all integral polyno- mials over a Banach space are extendible. Finally, we study the Aron-Berner extension of integral polynomials, and give an equivalence for non-containment of ℓ 1 . Introduction A natural question concerning scalar-valued continuous homogeneous polynomi- als over a Banach space E is whether they can be extended to a larger space G, much as linear forms can be extended by using the Hahn-Banach theorem. It is well known that the answer to this question is in general, no. There are several positive answers for particular situations. Most of these rely –explicitly or not– on the existence of a linear extension morphism for linear functionals E ′ −→ G ′ . This is the case of G = E ′′ ([3], [9]), of more general but similar constructions ([15], [27]), and even of the ultrapower methods employed by [13] and by [20], which can be seen to be related to the existence of such an extension morphism. This is of course stronger than the Hahn-Banach theorem, and is in fact equivalent to E ′′ being complemented in G ′′ (see also [19] and [21]). Moreover, it is equivalent to the existence of a linear extension morphism for continuous homogeneous polynomials P ( k E) −→ P ( k G). Here we are concerned with true Hahn-Banach type extensions; we want sufficient conditions for the extension of each P , not a linear morphism that will extend all of them. Our approach consists of identifying polynomials with linear functionals and us- ing the Hahn-Banach theorem to extend these. Thus in §1 we study preduals of several different spaces of polynomials. Note that we are only interested in identi- fying polynomials with linear forms, so our identifications of spaces of polynomials with spaces of linear forms need only be algebraic. It is important however that all polynomials be identified with linear forms over the same space, and that the different classes of polynomials appear as we vary the topology over this space. In a recent paper, Kirwan and Ryan [18] study the space of all “extendible” polynomials (i.e. those which can be extended to any larger space). We prove in Received by the editors September 5, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 46G20; Secondary 46B99. Key words and phrases. Extension of polynomials, containment of ℓ 1 . c 1999 American Mathematical Society 241 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use