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 Hn with dimHn→∞ and dimHn→∞ and lim
\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.