Characterization of nets with no convergent subnets in Banach spaces

Question

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?

Answer 1

As I understand, your question is about the equivalence for infinite dimensional Banach spaces of the two following statements:

a. $(x_{n}) \mbox{ in } X$ admits a convergent subsequence

b. $\displaystyle\liminf_{n} \|x_{n}\|<\infty$

If so, the answer is no. Certainly, in the finite dimensional case a. $\Leftrightarrow$ b.. In the infinite dimensional case, a. $\Rightarrow$ b. but the converse is not true. Here a counter example: in $\ell_{2}$ take the sequence of the unitary vectors $(e_{n})$; we have $\liminf \|e_{n}\|=1<\infty$ but it does not admit any norm convergent subsequence because being $(e_{n})$ a weakly null sequence if a subsequence $(e_{n_{k}})$ were norm convergent it should be convergent to zero in the norm and this is not the case since $\|e_{n_{k}}\|=1$.