Polynomial continuity on $\ell _1$

Abstract:

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. A Banach space $X$ has property (RP) if given two bounded sequences $(u_j), (v_j)\subset X$, we have that $Q(u_j)-Q(v_j)\rightarrow 0$ for every polynomial $Q$ on $X$ whenever $P(u_j-v_j)\rightarrow 0$ for every polynomial $P$ on $X$; i.e., the restriction of every polynomial on $X$ to each bounded set is uniformly sequentially continuous for the weak polynomial topology. We show that property (RP) does not imply that every scalar valued polynomial on $X$ must be polynomially continuous.