Polynomially reflexive space

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 "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.

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.

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.)

Thus if space contains $ell^p$ as a quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.

The symmetric Tsirelson space is polynomially reflexive.