Let $F$ be the real sequence which satisfies $ \sup _{n} {|nx_n|}<\infty$ in $\ell^2$. Show that
- $F$ are linear subspace of $\ell^2$.
- $F$ is closed in $\ell^2$.
My idea:
- is trivial.
- As is known to all, $\ell^2$ is a Banach space. The subspace of a Banach space is closed if and only if it is complete (Is this right?).
So we only to prove $F$ is complete subspace. But how can I show that every Cauchy sequence converge in $F$? Or we need to show that the closure of $F$ is itself? Could someone give me some details. Thank you!