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cabin5
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Homework Statement
Prove that the normed linear space [tex]l_{\infty}^{2}[/tex] is not an inner product space.
Homework Equations
parallelogram law;
[tex]\left\|x+y\right\|^2+\left\|x-y\right\|^2=2\left\|x\right\|^2+2\left\|y\right\|^2[/tex]
The Attempt at a Solution
Well, I tried to apply parallelogram law to the [tex]l_{\infty}^{2}[/tex] space where
[tex]x=(\alpha^1,\alpha^2)[/tex] and [tex]y=(\beta^1,\beta^2)\in l_{\infty}^{2} [/tex] .
[tex]\left\|x\right\|=max\left\{\left|\alpha^1\right|,\left|\alpha^2\right|\right\} and
\left\|y\right\|=max\left\{\left|\beta^1\right|,\left|\beta^2\right|\right\} [/tex]
If one puts these norms into the parallelogram law equation, one gets a fuzzy expression on both sides of the equation, therefore it is important to put out expressions inside the max{} function which I could not achieve to do.
Is there another method to solve this problem or am I misapplying the law to [tex]l_{\infty}^{2}[/tex] space?
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