We define the space Pn as consisting of all n-homogeneous polynomials.
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 xn satisfying g(xn) → 0 for all linear functionals g, satisfies also f(xn) → 0 for all quadratic forms f. In infinite dimension the situation is different.
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:[1] Quadratic forms are 2-homogeneous polynomials.
The equivalence mentioned above holds also for n-homogeneous polynomials, n=3,4,... For the
This makes polynomially reflexive spaces rare.