Question about the simple example for batch normalization given in "deep learning" book
I think what the statement meant was when given weights $w_1,...,w_n$ are fixed, output is linear proportional to $x$, but as it mentions


  nonlinear function of the weights w_i


Given a set of weights (more than one being varied), they do not linearly add to produce an output like

$y=w_1x_1+...+w_nx_n$

but rather non-linearly like

$y=w_1w_2..w_n*x$ (each $w_i$ is a dimension in the hyperspace)

And I think it would become more clearer from the statement "Output is linear to any weight $w_i$ but non-linear to weights $w_i$".