structural:reformatting, stylistic:tone
Properties of ℓ spaces

A sequence of elements in ℓ converges in the space of complex sequences ℓ if and only if it converges weakly in this space. If K is a subset of this space, then the following are equivalent:

Here K being equismall at infinity means that for every ε > 0 {\displaystyle \varepsilon >0}, there exists a natural number n ε ≥ 0 {\displaystyle n_{\varepsilon }\geq 0} such that ∑ n = n ϵ ∞ | s n | < ε {\textstyle \sum _{n=n_{\epsilon }}^{\infty }|s_{n}|<\varepsilon } for all s = (s n) n = 1 ∞ ∈ K {\displaystyle s=\left(s_{n}\right)_{n=1}^{\infty }\in K}.