# $c_0(N)$ is not dense in $c(N)$ for any norm.

If $$c(\mathbb{N})$$ is the space of all real-valued convergent sequences, $$c_0(\mathbb{N})$$ the subspace of sequences converging to $$0$$, I want to show that there is no norm so that $$c_0$$ is dense in $$c$$. I can show that $$c_0$$ is closed in the infinity norm, but not all norms are equivalent on $$c_0$$.

My idea is to show that $$c_{00}$$ is dense in $$c_0$$ for any norm, so if $$c_0$$ is dense in $$c$$ for some norm, so is $$c_{00}$$. I'm stuck trying to derive a contradiction from this, and I would appreciate any ideas!

## 1 Answer

I do not think that you can prove this proposition for all norms.

For a $$x=(x_0,x_1,...)\in c(\mathbb{N})$$ define the following norm $$||x||:=\sum_{k=0}^\infty \frac{|x_k|}{2^k}$$

Then we can approximate $$x\in c(\mathbb{N})$$ by a sequence $$y_j \in c_{00}(\mathbb{N})$$ via setting $$y_j :=(x_0,...,x_j,0,0,0....)$$.

• @MaoWao, convergent $\implies$ bounded. – Martín-Blas Pérez Pinilla Mar 9 '19 at 20:27
• Sorry, I misread it as all sequences. You are of course right. I will delete the misleading comment. – MaoWao Mar 10 '19 at 9:33