Proof of a normed space

cabin5

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.