Understanding a conclusion through "elementary considerations about bilinear forms"

Question

I am having a hard time to understand how "elementary considerations about bilinear forms" can imply the following result:

Let $E$ be a function space, 1 the constant function $x\mapsto 1$ and let $Q:E\times E\to\Bbb R$ be a bilinear form such that $Q(1,1)=0$ and $Q(\phi,\phi)\geq0$ for all $\phi\in E$. Then $Q(1,\phi)=0$ for all $\phi\in E$.

The reference is the paper Rigidity of Area-Minimizing Free Boundary Surfaces in Mean Convex Three-Manifolds by Lucas C. Ambrozio, page 6:

Answer 1

Since $Q$ seems symmetrical, from CS we have : $$ |Q(1,\phi)|^2 \leq Q(1,1)Q(\phi,\phi) = 0 $$ which implies $Q(1,\phi)=0$.