Polynomials and Identities on Real Banach Spaces

In our present paper we study the duality theory and linear identities for real polynomials and functions on Banach spaces, which allows for a unified treatment and generalization of some classical results in the area. The basic idea is to exploit point evaluations of polynomials, as e.g. in [Rez93]. As a by-product we also obtain a curious generalization of the well-known Hilbert lemma on the representation of the even powers of the Hilbert norm as sums of powers of functionals (Corollary 2.14). In Theorems 2.16 and 2.20 (generalizing [Wil18] and [Rez79]) we prove that identities derived from pairwise linearly independent point evaluations can be satisfied only by polynomials. We apply the Lagrange interpolation theory in order to create a machinery allowing the creation of linear identities which characterize spaces of polynomials of prescribed degrees (Theorem 2.18, Theorem 3.2). We elucidate the special situation when all the evaluation points are collinear (Corollary 2.24 and Theorem 3.4). Our work is based on (and generalizes) the theory of functional equations in the complex plane due to Wilson [Wil18] and Reznick (in the homogeneous case) [Rez78], [Rez79], the classical characterizations of polynomials due to Fréchet [Fr09], [Fr09b], and Mazur and W. Orlicz, [MO1], [MO2] which can be summarized in the following theorem.