ABSTRACT
4.1. Polynomials are the simplest holomorphic functions which are used to approximate all other holomorphic mappings; thus their properties should be investigated first. We define a homogeneous polynomial as the restriction of a symmetric multilinear map to the diagonal. In this chapter, the multinomial and polarization formulae are developed, several necessary and sufficient conditions for a polynomial to be continuous are studied, and the Banach-Steinhaus theorem is extended to the space of m-homogeneous polynomials.
