= Symmetry and coercivity in quadratic energy minimization
{title2=$I(v)=a(v,v)-2\ell(v)$}
A continuous symmetric <coercive bilinear form> $a$ on a real <Hilbert space> and a continuous <linear functional> $\ell$ give a unique minimum of $I$. The parallelogram identity makes any minimizing sequence Cauchy in the energy norm; coercivity and completeness give its limit. Differentiation gives $a(u,w)=\ell(w)$, and $I(v)-I(u)=a(v-u,v-u)$ proves uniqueness. A nonsymmetric operator instead differentiates to its symmetric part. Mere pointwise positivity without a lower bound in the chosen norm need not yield a minimizing vector.
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