Function values are enough for $L_2$-approximation: Part II

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.