Basis/Dual basis/Tautological lemma/Fact/Proof

The vector ${\displaystyle {}v}$ has a unique representation

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

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

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