Functional equations for polynomials

Abstract

Abstract. The set of all continuous symmetric multilinear forms of degree m on a real topological vector space V are shown to be in one-to-one correspondence with the family of continuous scalar-valued functions on V satisfying a certain functional equation. If V is «-dimensional, these functions are precisely those which can be represented by m-homogeneous polynomials of degree n (with respect to some basis of V).The connection between this family of generalized polynomials and the mth derivatives of a scalar-valued function is discussed.Denote by N the set of natural numbers and, for each n E N, by n the set [m E N: m < n). For all m, n E N, write n-\ for the family of increasing functions in n~.Let A be an Abelian group and B a linear space over a field of characteristic 0. The binary operations in A and B will both be denoted + and, for each n E N and x E A u B, nx will signify x + x + • • • + x (n times). For each m E N, smm(A, B) will be the set of all symmetric functions <b\A --> B such that, whenever k E m and x,y, z E A -satisfy xk + yk = zk and Xj = y} = Zj for ally ^ k, then <¡>(x) + <¡>(y) = <¡>(z).