Holomorphy Theory for Dual Pairs of Vector Spaces

Abstract

Let E be a complex convex linear topological space (CLTS) and denote by E^m the cartesian product E ×⋯× E (m factors). A function P: E → ℂ is called a homogeneous polynomial of degree m on E if there exists a nonzero, symmetric, m-linear functional

$$ \beta :{E^m} \to \mathbb{C},\quad ({x_1},...,{x_m}) \to \beta ({x_1},...,{x_m}) $$

such that

$$ P(x) = \beta (x,...,x),\quad x \in E $$

. A constant function is a homogeneous polynomial of degree zero. A function which is a finite sum of homogeneous polynomials is called a generalized polynomial. The collection of all generalized polynomials is obviously a point separating algebra of functions on E. Each of its elements P has a unique representation of the form P = P₀ +…+ P_m, where P_k is either zero or a homogeneous polynomial of degree k, for 0 ≤ k ≤ m, and P_m ≠ 0. The integer m is called the degree of P and the polynomials P_k are called its homogeneous components. If x₀ is a fixed element of E then Q(x) = P(x - x₀), x ∈ E, defines a generalized polynomial with degree equal to that of P. Observe that a linear functional on E is a polynomial of degree one. Elements of the algebra generated by the linear functionals are called finite polynomials.