Polynomially reflexive space


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 Mn of degree n, we can define a polynomial p as
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.

Relation to continuity of forms

On a finite-dimensional linear space, a quadratic form x↦f is always a linear combination of products x↦g h of two linear functionals g and h. Therefore, assuming that the scalars are complex numbers, every sequence xn satisfying g → 0 for all linear functionals g, satisfies also f → 0 for all quadratic forms f.
In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence xn satisfies g → 0 for all linear functionals g, and nevertheless f = 1 where f is the quadratic form f = ||x||2. In more technical words, this quadratic form fails to be weakly sequentially continuous at the origin.
On a reflexive Banach space with the approximation property the following two conditions are equivalent:
Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for n-homogeneous polynomials, n=3,4,...

Examples

For the spaces, the Pn is reflexive if and only if <. Thus, no is polynomially reflexive.
Thus if a Banach space admits as a quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare.
The Tsirelson space T* is polynomially reflexive.