Basis/Dual basis/Tautological lemma/Fact/Proof

Proof

The vector ${}v$ has a unique representation

${}v=\sum _{j=1}^{n}s_{j}v_{j}\,$

with ${}s_{j}\in K$. The right hand side of the claimed equality is therefore

${}\sum _{i=1}^{n}v_{i}^{*}(v)v_{i}=\sum _{i=1}^{n}v_{i}^{*}{\left(\sum _{j=1}^{n}s_{j}v_{j}\right)}v_{i}=\sum _{i=1}^{n}s_{i}v_{i}=v\,.$

To fact