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 "M" of degree "n" (that is, "M" is "n"-linear), we can define a polynomial "p" as (that is, applying "M" 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" as consisting of all "n"-homogeneous polynomials. The "P" 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"("x") is always a (finite) linear combination of products "x"↦"g"("x") "h"("x") of two linear functionals "g" and "h". Therefore, assuming that the scalars are complex numbers, every sequence "x" satisfying "g"("x") → 0 for all linear functionals "g", satisfies also "f"("x") → 0 for all quadratic forms "f". In infinite dimension the situation is different. For example, in a Hilbert space, an orthonormal sequence "x" satisfies "g"("x") → 0 for all linear functionals  g, and nevertheless "f"("x") = 1 where "f" is the quadratic form "f"("x") = ||"x"||. 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 formula_2 spaces, the "P" is reflexive if and only if < . Thus, no formula_2 is polynomially reflexive. (formula_4 is ruled out because it is not reflexive.) Thus if a Banach space admits formula_2 as a  quotient space, it is not polynomially reflexive. This makes polynomially reflexive spaces rare. The Tsirelson space "T"* is polynomially reflexive.