= Solution
Every <finitely supported sequence> $v$ lies in $Y$: if $v_n=0$ for $n>N$, set $x_n=\sum_{k=n}^Nv_k$ for $n\leq N$ and $x_n=0$ afterwards. Then $x$ is a <bounded sequence> and $x-Sx=v$. Consequently $L(v)=0$. The <finitely supported sequences> are dense in the <space of sequences converging to zero> $c_0$ for the <supremum norm>, by truncation, so continuity of the <bounded linear functional> $L$ gives $L|_{c_0}=0$. For $x$ in the real <convergent sequence space>, with ordinary limit $\ell$, we have $x-\ell\mathbf1\in c_0$. Therefore \b[$\boxed{L(x)=\ell L(\mathbf1)=\lim_{n\to\infty}x_n}$].
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