  In the first part we have shown that, for $L_2$-approximation of functions
from a separable Hilbert space in the worst-case setting, linear algorithms
based on function values are almost as powerful as arbitrary linear algorithms
if the approximation numbers are square-summable. That is, they achieve the
same polynomial rate of convergence. In this sequel, we prove a similar result
for separable Banach spaces and other classes of functions.
