Convex functional

A functional on a real vector space is convex when it is a convex function: $J(\theta v+(1-\theta )w)\le \theta J(v)+(1-\theta )J(w)$ for $0\le \theta \le 1$. For a symmetric nonnegative bilinear form $a$ and a linear functional $\ell$, the quadratic functional $J(v)=\frac{1}{2}a(v,v)-\ell (v)$ is convex, since its convexity difference is $\frac{1}{2}\theta (1-\theta )a(v-w,v-w)\ge 0$.