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

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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.

asked Jul 7, 2017
edited Jul 8, 2017

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