Editing Polynomially reflexive space (section)

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 map|multilinear]] [[functional (mathematics)|functional]] ''M''<sub>''n''</sub> of degree ''n'' (that is, ''M''<sub>''n''</sub> is ''n''-linear), we can define a polynomial ''p'' as

:<math>p(x)=M_n(x,\dots,x)</math>

(that is, applying ''M''<sub>''n''</sub> 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''<sub>''n''</sub> as consisting of all ''n''-homogeneous polynomials.

The ''P''<sub>1</sub> is identical to the [[dual space]], and is thus reflexive for all reflexive ''X''. This implies that reflexivity is a prerequisite for polynomial reflexivity.