I have been reading about the $\ell_0$ norm, wikipedia gives us that "The mathematical definition of the $\ell_0$ norm was established by Banach's Theory of Linear Operations. The space of sequences has a complete metric topology provided by the F-norm $$(x_n) \mapsto \sum_{n=0}^{\infty} \frac{1}{2^n} \frac{\left|x_n \right|}{1+\left| x_n \right|}.$$ Which spaces are complete with respect to this norm? The set of all bounded sequences, the sequences consisting only of zeros and ones? Any more?
$$\| \lambda x \| = \sum_{n=1}^{\infty} 2^{-n} \frac{\left| \lambda x_n \right|}{1+\left| \lambda x_n \right|} \leq \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{\left| \lambda \right| \left| x_n \right|}{1+\left| \lambda \right| \left| x_n \right|} .... $$ Where do we go from here?
As John Ma said, this "norm" is not norm. To avoid saying things that aren't really true, it's better to talk about the $\ell_0$ metric $$ d_0((x_n),(y_n)) = \sum_{n=0}^{\infty} \frac{1}{2^n} \frac{\left|x_n-y_n \right|}{1+\left| x_n-y_n \right|}. $$ This is a metric on the space of all sequences, which turns it into a complete metric space $S_0$. The convergence with respect to this metric is precisely coordinate-wise convergence.
A sequence space is complete with respect to $d_0$ if and only if it is a closed subset of the metric space $S_0$. So the answer to your question is: a sequence space is complete with respect to $d_0$ if and only if it is closed under coordinate-wise convergence.
The set of all bounded sequences is not complete. Consider the sequence of sequences $(x_n^{(m)})$ where $x_n^{(m)} = \min(m,n)$. For each $m$ this is a bounded sequence. It is Cauchy with respect to $d_0$, because it converges in $S_0$. But its limit is an unbounded sequence.
The sequences consisting only of zeros and ones is complete. So is the space of sequences taking values only in a given closed set $A\subset\mathbb{R}$. This is a lot of examples. One can come up with other variations: require every even-numbered term to be $0$, or to be twice the value of the preceding odd-numbered term, etc.