prove unicity of a linear continous form

Question

Let H be a Hilbert space and $ u ∈ H$.

Prove there exists a unique continuous linear form $L∈ H^*$ such that: $||L||_{H^∗} = ||u||_H$ and $<L, u> = ||u||^2_H$

I proved the existence : We can take $<L, x> = ||u||_H||x||_H$, but I don't have a clue about unicity.

Thank you for your help.