The role of $dim H_n$ in the definition of asymptotically continous functions on vectors

Question

When considering the asymptotic continuity of quantum states, one works with asymptoticly continuous functions.

In the definition one has the following, a funtion f is asymptotically cts if for a family of hilbert spaces $H_n$ with $dim H_n\rightarrow\infty$ and $\dim H_n\rightarrow\infty$ and $\lim\limits_{n\rightarrow\infty}\rho_n = \lim\limits_{n\rightarrow\infty}\sigma_n$

$\lim\limits_{n\rightarrow\infty} \frac{||f(\rho_n)-f(\sigma_n)||}{dim H_n}= 0$

where n is in the natural numbers $\rho_n$ and $\sigma_n$ are states in a sequence indexed by n, and anything I've missed has the usual definition.

My question is what is the point of $H_n$ with $dim H_n\rightarrow\infty$ why not simply say when the function on the sequence of states gets arbitarily close so does the difference between the asymptotically cts functions on them. I intend to abstract this definition, and it seems to work fine, however the existence of this dimension term on the bottom for the particular case of vector spaces worries me.