The Auerbach Lemma
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The Auerbach Lemma
Let \(\displaystyle{\left(X,||\cdot||\right)}\) be a finite-dimensional normed space with \(\displaystyle{\dim_{\mathbb{R}}X=n\in\mathbb{N}}\) .
Prove that there are \(\displaystyle{x_1,...,x_n\in X}\) and \(\displaystyle{f_1,...,f_n\in X^{\star}}\)
such that \(\displaystyle{||x_i||=||f_{i}||=1\,,\forall\,i\in\left\{1,...,n\right\}}\) and
\(\displaystyle{f_{i}(x_{j})=\delta_{i\,j}\,,\forall\,i\,,j\in\left\{1,...,n\right\}}\) .
Prove that there are \(\displaystyle{x_1,...,x_n\in X}\) and \(\displaystyle{f_1,...,f_n\in X^{\star}}\)
such that \(\displaystyle{||x_i||=||f_{i}||=1\,,\forall\,i\in\left\{1,...,n\right\}}\) and
\(\displaystyle{f_{i}(x_{j})=\delta_{i\,j}\,,\forall\,i\,,j\in\left\{1,...,n\right\}}\) .
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