Which of the following spaces are dense?

Question

Let $X$ be the set of all real sequences, consider the subset $$S=\big\{x=\{x_n\}: x_n\in\mathbb{Q}~\forall n, x_n=0\text{ except for a finite number of n}\big\}.$$

We need to find out which of the followings are true.

• $a.$ $S$ is dense in $l_1$, the space of absolutely summable sequences, provided with the metric $$d_1(x,y)=\sum_{n=1}^{\infty}|x_n-y_n|\;.$$

• $b.$ $S$ is dense in $l_2$, the space of square summable sequences, provided with the metric $$d_2(x,y)=\left(\sum_{n=1}^{\infty}|x_n-y_n|^2\right)^{1/2}.$$

• $c.$ $S$ is dense in $l_{\infty}$, the space of bounded sequences, provided with the metric $$d_{\infty}(x,y)=\sup_n\{|x_n-y_n|\}\;.$$

My Intution: I know that $l_p$ is seperable for $1\le p<\infty$, as $S$ is countable and dense so $a,b$ are true, $c$ is false as $l_{\infty}$ is not seperable. Am I right?

Answer 1

Separability of $\ell_p$ by itself is not sufficient to state that some countable set is dense, but gives you a hint on how to construct an approximation sequence in $S$ for any sequences in $\ell_p$ when $p<\infty$. For the case $c$ consider the sequence consisting of $1$'s.