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

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

(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 weak sequentially continuous.

Examples

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

Thus if space contains <math>\ell^p</math> as a quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.

The symmetric Tsirelson space is polynomially reflexive.