Polynomially Reflexive Space

In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space.

Given a multilinear functional M_n of degree n (that is, M_n is n-linear), we can define a polynomial p as

(that is, applying M_n on the diagonal) or any finite sum of these. If only n-linear functionals are in the sum, the polynomial is said to be n-homogeneous.

We define the space P_n as consisting of all n-homogeneous polynomials.

The P₁ is identical to the dual space, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.