I read the Lax-Milgram Theorem in the *Navier-Stokes Equations* by Temam:

Let $X$ be a

separableHilbert space (norm $\|\cdot\|_X$) and let $$ a:X\times X\to\Bbb{R} $$ be a bilinear continuous coercive form; that is, there exist $c,C>0$, such that for all $u,v\in X$, we have \begin{align} |a(u,v)|&\leq C\|u\|_X\|v\|_X&\text{(continuous)}\\ a(u,u)&\geq c\|u\|_X^2 &\text{(coersive)} \end{align} Then, for each continuous functional $\lambda$ on $X$, there exists a unique element $u\in X$ such that $$ a(u,v)=\langle \lambda,v\rangle\quad\text{for all }v\in X. $$

The proof of existence in his book explicitly exploits the separability of the space. Some people define Hilbert spaces with the separability assumption. Is this theorem true without the separability assumption? (If yes, would you come up with some cited references?)