Let $X$ be a finite-dimensional Banach space and $(x_i)_{i\in I}$ a net in $X$. Since every limited net in $X$ has a convergent subnet, it follows that $(x_i)_{i\in I}$ does not admits a convergent subnet if and only if $$ \liminf_{i\in I} \|x_i\|=+\infty.$$

I wonder if there is such a characterization (by means of the norms or differences of the $x_i$'s) for some class of infinite dimensional Banach spaces.

Is there a paper, survey or even a chapter of a book containing any properties on this type of net?