Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace consisting of finite sums of elements from $\cup_n F_n$. Do there exist criteria for telling when the topological closure $\overline{\sum_n F_n}$ is the Banach space sum of the $F_n$'s? By that I mean that every element of $\overline{\sum F_n}$ can be written in a unique way as an absolutely convergent sum $\sum_{n=1}^\infty f_n$ $(f_n\in F_n)$.
I'm particularly interested in the case where $F_n =E_n$ in this post https://math.stackexchange.com/questions/1210693/a-question-about-a-proof-in-langs-sl-2-mathbbr. So $$F_n = \{v\in E : \pi(r_\theta)v=e^{in\theta}v \hspace{3mm} \forall r_\theta \in SO_2(\mathbb{R})\}$$ where $\pi$ is some irreducible strongly continuous representation of $SL_2(\mathbb{R})$ in a Banach space $E$.