I recently started reading Filipovic's Consistency problems for HJM interest rate models and came across the Musiela reparametrization
$$r_t(x)=f(t,x+t)$$ so the forward curve can be thought of as a map $x\to r_t(x)$. The book goes on to tell that this maybe thought of as an infinite dimensional state variable.
Does anyone have a good explanation for this? Is it because we pick $r_t$ from a space of functions?