Polynomially reflexive space

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In mathematics, a polynomially reflexive space is a Banach space X, on which all polynomials are reflexive.

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

$p(x)=M_n(x,\dots,x)$

(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.

In the presence of the approximation property of X, a reflexive Banach space is polynomially reflexive, if and only if every polynomial on X is weakly sequentially continuous.

[edit] Examples

For the $\ell^p$ spaces, the P_n is reflexive if and only if n < p. Thus, no $\ell^p$ is polynomially reflexive. ( $\ell^\infty$ is ruled out because it is not reflexive.)