Notes on the Geometry of Space of Polynomials

The real geometric properties of spaces of polynomials are discussed in [1, 6]. In particular, it is shown that the symmetric injective tensor product space ⊗̂n,s,εE is not strictly convex if E is a Banach space of dimE ≥ 2 and if n ≥ 2 holds. Let E be a Banach space over a real or complex filed and E is denoted as the Banach dual of E. An element x in the unit sphere SE is called a (real) extreme point of the unit ball BE if for some y ∈ E, ‖x± y‖ ≤ 1 implies y = 0. Recall that a Banach space E is said to be strictly convex if every element of SX is an extreme point of BE . Suppose for the moment that E is a complex Banach space. An element x in the unit sphere SE is said to be a complex extreme point if for some y ∈ E, sup{‖x+ ζy‖ : ζ ∈ C, |ζ| = 1} ≤ 1 implies y = 0. A complex Banach space E is said to be complex strictly convex if every point in SE is a complex extreme point of BE . Notice that if a complex Banach space is not complex strictly convex, then it is not strictly convex. Given a Banach space E, the space ⊗n,sE consists of all tensors of the form