Double sums of vectors in a Banach space

Question

Let $E$ be a Banach space of infinite dimension (you can assume that $E=c_0$ if it is necessary), let $(\alpha_i)_{i\in\mathbb{N}}$ be a sequence of scalars and let $(z_k)_{k\in\mathbb{N}}$ be a sequence of vectors in $E$. Suppose that the following four series are convergent $$\sum_{i=1}^\infty \alpha_i\;, \sum_{k=1}^\infty z_k\;, \sum_{i=1}^\infty\left(\sum_{k=1}^\infty \alpha_iz_k\right), \mbox{ and } \sum_{k=1}^\infty\left(\sum_{i=1}^\infty \alpha_i\right)z_k.$$

Question: Does the following equality holds? $$\sum_{i=1}^\infty\left(\sum_{k=1}^\infty \alpha_iz_k\right)= \sum_{k=1}^\infty\left(\sum_{i=1}^\infty \alpha_i\right)z_k.$$

Answer 1

Write $\alpha=\sum_i\alpha_i$, $z=\sum_kz_k$. Then $$\sum_{i=1}^\infty\left(\sum_{k=1}^\infty\alpha_iz_k\right)=\sum_{i=1}^\infty \left(\alpha_i\left(\sum_{k=1}^\infty z_k\right)\right)=\sum_{i=1}^\infty \alpha_iz=\left(\sum_{i=1}^\infty\alpha_i\right)z=\alpha z$$ and $$\sum_{k=1}^\infty\left(\sum_{i=1}^\infty \alpha_i\right)z_k=\sum_{k=1}^\infty\alpha z_k=\alpha\sum_{k=1}^\infty z_k=\alpha z.$$