 Solution

Yes. For $x=(x_n)\in c$, let $L={\lim }_n{x}_n$ and defineThe tail tends to zero, so $T(x)\in c_0$, and $\Vert T(x){\Vert }_{\infty }\le 2\Vert x{\Vert }_{\infty }$. Conversely, for $y=(y_n)\in c_0$, defineThis sequence converges to $y_1$ and has norm at most $2\Vert y{\Vert }_{\infty }$. The formulas are inverse bounded linear maps, so $c$ and $c_0$ are isomorphic Banach spaces.