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 Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as
- <math>p(x)=M_n(x,\dots,x)<math>
(that is, applying Mn 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 Pn 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 Pn 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.