# Monotonicity of the norms on the sequence spaces 2

This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part).

Let $$\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty]$$ satisfy the following properties:

• $$\rho(\lambda u)=\lambda\rho( u)$$, for every $$u\in [0,+\infty)^{\mathbb{N}}$$ and $$\lambda\ge 0$$;

• If $$\rho( u)=0$$, then $$u=0_{\mathbb{R}^{\mathbb{N}}}$$;

• $$\rho(u+v)\le \rho( u)+\rho( v)$$, for every $$u,v\in [0,+\infty)^{\mathbb{N}}$$.

• $$\rho(\{u_n\}_{n\in\mathbb{N}})\le \rho(\{v_n\}_{n\in\mathbb{N}})$$, once $$0\le u_n\le v_n$$, for all $$n$$, and the set $$\{\frac{u_n}{v_n}, n\in \mathbb{N}\}$$ is finite (we count $$\frac{0}{0}=0$$).

Define $$\|\cdot\|:\mathbb{R}^{\mathbb{N}} \to[0,+\infty]$$ by $$\left\|\{u_n\}_{n\in\mathbb{N}}\right\|=\rho(\{|u_n|\}_{n\in\mathbb{N}})$$.

Does it follow that $$E=\{u\in \mathbb{R}^{\mathbb{N}}, \|u\|<+\infty\}$$ is a linear space and $$\|\cdot\|$$ is a norm on $$E$$?