Ritz-Galerkin equivalence for a symmetric coercive form ($J(v)-J(u)=\frac{1}{2}a(v-u,v-u)$)

For a bounded symmetric coercive bilinear form on a Hilbert space, a solution of $a(u,w)=\ell (w)$ minimizes $J(v)=a(v,v)/2-\ell (v)$. The displayed identity proves strict minimality and uniqueness. Restricting to a conforming finite element space makes the discrete Ritz method equivalent to the Galerkin method. Galerkin orthogonality gives exact best approximation in the energy norm; the Céa lemma translates it into a reference-norm estimate.

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.