- #1
cabin5
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Homework Statement
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]
Homework Equations
All conditions satisfied for a normed space.
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.