Local properties of polynomials on a Banach space

We introduce the concept of a smooth point of order n of the closed unit ba ll of a Banach space E and characterize such points for E = c(0), L-p(mu) ( 1 less than or equal to p less than or equal to infinity), and C(K). We sho w that every locally uniformly rotund multilinear form and homogeneous poly nomial on a Banach space E is generated by locally uniformly rotund linear functionals on E. We also classify such points for E = c(0), L-p(mu) (1 les s than or equal to p less than or equal to infinity), and C(K).