ON A NORMAL FORM FOR NON-WEAKLY SEQUENTIALLY CONTINUOUS POLYNOMIALS ON BANACH SPACES

Abstract

Let $p$ be an $m$-homogeneous polynomial on a complex Banach space, and let $(x_n)_n$ be a bounded sequence such that when evaluated in polynomials of degree less than $m$, it converges to zero, but $p(x_n)=1$. It is proved here that there exists a basic sequence $(y_k)_k$ equivalent to a subsequence $(x_{n_k})_k$, for which $p(\sum_{k=1}^{\infty}a_ky_k)=\sum_{k=1}^{\infty}a_k^m$.