which of the following metric spaces are separable?
$C[0,1]$ with usual 'sup norm' metric.
the space $l_1$ of all absolutely convergent real sequences, with the metric $d_1(a_i,b_i)=\sum_{1}^{\infty}|a_i-b_i|$
The space $l_{\infty}$ of all bounded real sequences with the metric $d_{\infty}(a_i,b_i)=\sup|a_i-b_i|$
Well, 1 is separable as polynomials are dense in $C[0,1]$ so I can construct a set of polynomial with rational coefficients that is going to be a countable dense set for $C[0,1]$
I have no idea about 2,3 .
Well, along with this question I just want to ask The closed unit ball is compact with respect to $l_1$ metric? I guess no, because Sequence $e_1=(1,0,\dots),\dots e_n=(0,0,\dots,1(nth place),0,0\dots)$ this seqquence has no convergent subsequence so not sequentially compact. Am I right?