= Inner product determined by a basis
{title2=$\langle u,v\rangle_e=\sum_i u_i\overline{v_i}$}
Each <basis> $e_1,\ldots,e_n$ of a finite-dimensional complex <vector space> determines a unique <complex inner product> making that <basis> <orthonormal>. In coordinates it is $\langle\sum_i u_ie_i,\sum_i v_ie_i\rangle_e=\sum_i u_i\overline{v_i}$, with the first argument linear. The formula is <sesquilinear>, has <conjugate symmetry>, and satisfies $\langle v,v\rangle_e=\sum_i|v_i|^2$, which is positive unless $v=0$. Expanding any <inner product> in the given <basis> proves uniqueness. Its codomain is the <complex numbers>; only its diagonal values are necessarily real. The resulting identification with the <dual space> is conjugate-linear, not complex-linear.
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