Question: In Berberian's Lectures in Functional Analysis and Operator Theory, I'm having trouble following the Aperitif on Wiener's Theorem and some of lapses in logic present within; one in particular.

On the proof of the convergence of $ \sum\_{n=-\infty}^{\infty} x(m - n)y(n) $, the author asserts the following equivalence:

$ \sum\_{m=-\infty}^{\infty} x(m - n) = \|x\|\_1 $ - note the change in index.

Given that x $ \in l^1(\mathbb{Z}) $ for $ x = \sum\_{n=-\infty}^{\infty} x(n) e\_n $ 

where $e\_n (m) = \delta\_{mn}$

and $(xy)(m)= \sum\_{n=-\infty}^{\infty} x(m - n)y(n) $ for $ m \in \mathbb{z}$

Why does $ \sum\_{m=-\infty}^{\infty} |x(m - n)|$ = $\|x\|\_1$ ?

Why does a "one" just magically appear as a subscript?

Note on edits: I had $y(n)$ originally multiplied to the sum, but that was not what I was intending to ask about; my apologies.

Answer: 
"Why does a "one" just magically appear as a subscript?" It's the definition of the $\ell^{1}(\mathbb{Z})$-norm. Given a sequence $a : \mathbb{Z} \to \mathbb{C}$, we define the norm $\|\cdot\|\_{1}$ by 
\begin{equation\*}
\|a\|\_{1} = \sum\_{n = - \infty}^{\infty} |a(n)|.
\end{equation\*}
Of course, this is only a norm on the vector space of sequences for which the sum is finite. This is precisely the definition of the Banach space $\ell^{1}(\mathbb{Z})$. 

Now given a sequence $x : \mathbb{Z} \to \mathbb{C}$ and an integer $n \in \mathbb{Z}$, substitution implies that 
\begin{equation\*}
\sum\_{m = - \infty}^{\infty} |x(m - n)| = \sum\_{m = - \infty}^{\infty} |x(m)| = \|x\|\_{1}.
\end{equation\*}
In case we are talking about numbers, clearly
$$
\sum\_{m=-\infty}^{\infty} |x(m - n)y(n)| = \|x\|\_1y(n),
$$
as it must be in the book (it cannot be as you wrote).