1. The problem statement, all variables and given/known data The norm is defined by [tex]\left\|x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}[/tex] where [tex]x=(\alpha^{1},\alpha^{2},....,\alpha^{n})[/tex] Prove that this is a norm on [tex]V_{\infty}(F)[/tex] 2. Relevant equations All conditions satisfied for a normed space. 3. The attempt at a solution Well, I proved the first condition which is i) [tex]\left\|x\right\|_{p}=\left\{\left|\alpha^{1}\right|^{p}+\left|\alpha^{2}\right|^{p}+.....+\left|\alpha^{n}\right|^{p}+....\right\}^{1/p}[/tex] This must be positive definitive, therefore [tex]\left\|x\right\|_{p}>0[/tex] On the second condition I don't know whether taking the product of this norm with a [tex]\beta\in F[/tex] since the sum is infinite. I got stuck at this point. And also I presume, for the third condition, Minkowski inequality cannot be used anymore to prove it.