Editing Polynomially reflexive space (section)

==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 functional]]s ''g'' and ''h''. Therefore, assuming that the scalars are complex numbers, every sequence ''x<sub>n</sub>'' satisfying ''g''(''x<sub>n</sub>'') &rarr; 0 for all linear functionals ''g'', satisfies also ''f''(''x<sub>n</sub>'') &rarr; 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>'' [[Weak convergence (Hilbert space)#Weak convergence of orthonormal sequences|satisfies]] ''g''(''x<sub>n</sub>'') &rarr; 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 [[Weak convergence (Hilbert space)|weakly]] [[Continuous function (topology)#Sequences and nets|sequentially continuous]] at the origin.

On a [[Reflexive space|reflexive]] [[Banach space]] with the [[approximation property]] the following two conditions are equivalent:<ref>Farmer 1994, page 261.</ref>
* every quadratic form is weakly sequentially continuous at the origin;
* the Banach space of all quadratic forms is reflexive.

Quadratic forms are 2-homogeneous polynomials. The equivalence mentioned above holds also for ''n''-homogeneous polynomials, ''n''=3,4,...