# Proof of a normed space

by cabin5
Tags: normed, proof, space
 P: 18 1. The problem statement, all variables and given/known data The norm is defined by $$\left\|x\right\|_{p}=\left[\sum^{\infty}_{k=1}\left|\alpha^{i}\right|^{p}\right]^{1/p}$$ where $$x=(\alpha^{1},\alpha^{2},....,\alpha^{n})$$ Prove that this is a norm on $$V_{\infty}(F)$$ 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) $$\left\|x\right\|_{p}=\left\{\left|\alpha^{1}\right|^{p}+\left|\alpha^{2 }\right|^{p}+.....+\left|\alpha^{n}\right|^{p}+....\right\}^{1/p}$$ This must be positive definitive, therefore $$\left\|x\right\|_{p}>0$$ On the second condition I don't know whether taking the product of this norm with a $$\beta\in F$$ 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.

 Related Discussions Linear & Abstract Algebra 1 Calculus & Beyond Homework 2 Calculus & Beyond Homework 2 Calculus & Beyond Homework 1 Calculus & Beyond Homework 22