Given $C=\{e_1^*,e_2^*,...,e_n^*\}$, the canonical basis of $(R_n)^*$ (dual of $R_n$). I need to prove that:
$\forall i \in I_n=\{1,...,n\}, \forall u \in R^n: <e_i^*,u>=e_i^tu$
I can use the Riesz-Fréchet representation theorem to prove it but I am not sure how. Can I get some help/ideas? Thanks.
Recall that the basis $C$ acts as on the canonical basis $e_1, \ldots e_n$ as $e_i^*(e_j) = \delta_{ij}$.
Hence we have $$e_i^*\left(u\right) = e_i^*\left(\sum_{j=1}^n u_ie_i\right) = \sum_{j=1}^n u_j e_i^*(e_j) = \sum_{j=1}^n u_j \delta_{ij} = \sum_{j=1}^n u_j (e_i)_j = e_i^tu$$