Let $\ell^2$ denote the space of all square-summable real sequences.
Let $\{c_k\}$ be a fixed sequence of real numbers. Define an operator $D$ on $\ell^2$ by $D\{x_k\} = \{c_kx_k\}$.
Suppose that $D$ is a well-defined operator mapping $\ell^2$ into $\ell^2$ continuously. Prove that $\sup_k c_k$ is finite.
I have tried to prove this but every time I try I end up with inequalities pointing in the wrong direction and it doesn't help.
I don't want a proof, but I would like a hint or two if possible.