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 *P*_{1} is identical to the dual space, and is thus reflexive for all reflexive *X*. This implies that reflexivity is a prerequisite for polynomial reflexivity.

Read more about Polynomially Reflexive Space: Relation To Continuity of Forms, Examples

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