Functional analysis and Banach Algebras well defined I have a question regarding well - definedness.
Suppose $X$ is a banach space $\mathcal{l}^{1}(\mathbb{Z})$ given by the norm $||(x_{n})_{n}||_{1} := \sum_{n \in \mathbb{Z}} |x_{n}|$
If we define the product $xy$ as $(xy)_{n} = \sum_{m} x_{m}y_{n-m}$.
Then is it correctly understood that showing
$\sum_{n} |(xy)_{n}| \leq ||x|||y||$ and that $X$ has a unit implies that
$(xy)_{n} = \sum_{m} x_{m}y_{n-m}$ is a WELL DEFINED PRODUCT which makes $X$ a Banach algebra?

So first of all I show that the product is indead in $\mathcal{l}^{1}(\mathbb{Z})$
\begin{align*}
 \sum_{n} |(xy)_{n}| &\leq \sum_{n} \sum_{m} |x_{m}| |y_{n-m}|\\
 &= \sum_{m} |x_{m}| \sum_{n} |y_{n-m}|\\
 &= \sum_{m} |x_{m}| ||y||\\
 &= ||x|| \ ||y||
\end{align*}
which implies that $xy \in \mathcal{l}^{1}(\mathbb{Z})$
which shows that $\sum_{n} |(xy)_{n}| < \infty$.
Furthermore,
\begin{align*}
    x(y+z)_{n} &= x_{m} (\sum_{m} y_{n-m} + \sum_{m} z_{n-m})\\
               &= \sum_{m} x_{m} y_{n-m} + \sum_{m} x_{m} z_{n-m}\\
               &= (xy)_{n} + (xz)_{n}
\end{align*}
and
\begin{align*}
    ((x \cdot y) \cdot z)_{n} &= \sum_{m} (x_{m} y_{n-m}) z_{m})\\
                              &= \sum_{m} x_{m} (y_{n-m} z_{m})\\
                              &= (x(y \cdot z))_{n}
\end{align*}
which shows that the product is well defined
