# Null sequences with convergent series

What can be said topologically about the subspace $\{(a_k)_{k\ge1}\in \ell^\infty\mid\text{$\lim_{k\to\infty}a_k=0\wedge\sum_{k=1}^\infty a_k \ \ converges$}\}$ relative to the subspace $\{(a_k)_{k\ge1}\in \ell^\infty\mid\text{$\lim_{k\to\infty}a_k=0$}\}$ . Thanks

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You mean like, are they closed in $\ell^\infty$? – 1015 Feb 5 '13 at 19:51
@DavidMitra Are you sure? The second space is $c_0$, but the first is not closed in $l_\infty$, so it is not a Banach space. – Theo Feb 5 '13 at 22:11
@theo Ah, you're right, of course. I misread it... – David Mitra Feb 5 '13 at 23:24
BTW, convergence of the series $\sum a_k$ implies $a_k\to 0$. – Jochen Feb 6 '13 at 12:40

The second space is the classical space $c_0$ and is a closed subspace of $l_\infty$. The first space, call it $X$, is not closed in $l_\infty$. Indeed, the sequence $x_n=(1, 1/2, \dots, 1/n, 0, 0, \dots)$ is in $X$ and converges to $x=(1, 1/2, 1/3,\dots)$ which is not in $X$. However $X$ is a dense subspace of $c_0$. To see this, note that $c_{00}\subset X\subset c_0$ and the closure of $c_{00}$ in the $l_\infty$ norm is $c_0$. By $c_{00}$ we understand the subspace of finitely supported sequences in $c_0$.