Let $E$ be an infinite dimensional Banach space with Schauder basis $\{e_n:n\in\mathbb{N}\}$ and let $(w_n)_{n\in\mathbb{N}}$ be a sequence of vectors in $E$ such that, for each $n\in\mathbb{N}$, $||w_n||=1$ and there exists sequence of scalars $(\alpha_{i,n})_{i\in\mathbb{N}}$ satisfying $w_n=\sum_{i=1}^\infty\alpha_{i,n}e_i$. Let $(\beta_n)_{n\in\mathbb{N}}$ be a sequence of scalars such that $\lim_{n\to\infty}\beta_n=0$. Suppose that the following two series are convergent $$\sum_{n=1}^\infty \beta_n w_n=\sum_{n=1}^\infty \beta_n \left(\sum_{i=1}^\infty\alpha_{i,n}e_i\right)= \sum_{n=1}^\infty \left(\sum_{i=1}^\infty\beta_n \alpha_{i,n}e_i\right)\hspace{5mm}(1)$$ $$\mbox{ and }\sum_{i=1}^\infty \left(\sum_{n=1}^\infty\beta_n \alpha_{i,n}\right)e_i.\hspace{35mm}(2)$$
Question: Does the following equality holds? $$\sum_{n=1}^\infty \left(\sum_{i=1}^\infty\beta_n \alpha_{i,n}e_i\right)=\sum_{i=1}^\infty \left(\sum_{n=1}^\infty\beta_n \alpha_{i,n}\right)e_i.$$
A basic property of a Schauder basis is that coordinate functionals are continuous. Let $f_k$ be the coordinate functional for the $k$th vector. The continuity implies that
$$f_k\left(\sum_{n=1}^\infty \left(\sum_{i=1}^\infty\beta_n \alpha_{i,n}e_i\right)\right) = \sum_{n=1}^\infty f_k\left(\sum_{i=1}^\infty\beta_n \alpha_{i,n}e_i\right) = \sum_{n=1}^\infty\beta_n\alpha_{k,n}$$
And since $\sum f_k(x)e_k = x$ for any vector $x$, the conclusion follows.