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Proof of a normed space 
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#1
Sep1708, 09:19 AM

P: 18

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. 


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