HJM in infinite dimensions

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?

Keeping it simple, you know HJM SDE gives the dynamics of an instantaneous forward referencing a fixed maturity T, $$f\left(t, T\right)$$, but there is a continuum of such maturities - the whole forward curve as a function of T.
• Say we want to price a set of bonds with maturities $T$ varying on a certain interval so we have a continuum of bonds. In this case I see why we end up with a continuum of forwards. But what if we just take a finite set of bonds? Aug 25 '19 at 16:43
• But then even a single maturity zero coupon depends on the entire range of instantaneous forwards, remember the relationship: $B=e^{-\int_{0}^{T}{f(0,u)du}}$ Aug 25 '19 at 16:58