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<sub>n</sub> of degree n (that is, M<sub>n</sub> is n-linear), we can define a polynomial p as
(that is, applying M<sub>n</sub> 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<sub>n</sub> as consisting of all n-homogeneous polynomials.
The P<sub>1</sub> is identical to the dual space, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.
On a finite-dimensional linear space, a quadratic form xâ¦f(x) is always a (finite) linear combination of products xâ¦g(x) h(x) of two linear functionals g and h. Therefore, assuming that the scalars are complex numbers, every sequence x<sub>n</sub> satisfying g(x<sub>n</sub>) → 0 for all linear functionals g, satisfies also f(x<sub>n</sub>) → 0 for all quadratic forms f.
In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence x<sub>n</sub> satisfies g(x<sub>n</sub>) → 0 for all linear functionals g, and nevertheless f(x<sub>n</sub>) = 1 where f is the quadratic form f(x) = ||x||<sup>2</sup>. 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,...
For the spaces, the P<sub>n</sub> is reflexive if and only if < . Thus, no is polynomially reflexive. ( is ruled out because it is not 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.