# Coercive vs positive definite for a symmetric bilinear form

I was wondering once again what the difference in concept of positive definite and coercive was. Of course, if you have a bilinear form $a(.,.)$, that is coercive, positive definiteness follows directly from the definition: $$a(u,u)\geq\alpha\left\|u\right\|^2>0\text{ for }u\neq 0$$ since $\alpha>0$. Hope, I'm not wrong so far. On the other hand, I do have neither a proof nor an example that either proves or disproves the other direction. Is anyone here to clarify this?

On $L^2([0,1])$, consider $a(u,v)=\int_0^1 tu(t)\bar v(t)\,dt$. It is positive definite but not coercive.