Norm on dual space

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June 2nd 2011, 12:11 AM
geezeela

Norm on dual space

If X is the space of ordered n-tuples of real numbers and $||x||=max|{\xi}_{j}|$ where $x = ({\xi}_{1 }.......,{\xi}_{n})$ , what is the corresponding norm on the dual space X'?
June 2nd 2011, 01:16 AM
girdav

$x'\in X'$ can be represented by $(a_1,\ldots,a_n)\in\mathbb{R}^n$ (we have $\langle x',x\rangle =\sum_{j=1}^na_jx_j$ ). We have $\lVert x'\rVert=\sup_{\lVert x'\rVert =1} \left|\langle x',x\rangle\right|$ and taking $x_j = \mathrm{sgn}a_j$ we find $\lVert x'\rVert =\sum_{j=1}^n|a_j|$ .

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