Let $\ell^2(\mathbb{Z})$ be the set of all two-sided sequences $(a_i)$ in $\mathbb{C}$, such that $\sum_{n\in \mathbb{Z}} |a_n|^2 \lt \infty$.
What considerations do I have to take into account when separating sequences into two sums?
For example, an inner product on $\ell^2(\mathbb{Z})$ is defined for $a,b \in \ell^2(\mathbb{Z})$ by $(a,b) = \sum_{n\in\mathbb{Z}} a_n \overline{b_n}$ where the overline means complex conjugation. I want to prove that it's linear in the first variable, i.e. $(\alpha a + b, c) = \alpha(a,c) + (b,c). \ $ In doing so we have the sum $(\alpha a + b, c) = \sum (\alpha a_n + b_n)\overline{c_n}$. Why is it valid to separate the sum into two: $ = \alpha \sum a_n\overline{c_n} + \sum b_n\overline{c_n}$ ? I thought such operations on sequences were sometimes invalid. Thanks.