Lax-Milgram theorem states that
If $B(,)$ is a symmetric,strictly positive and bounded bilinear form on Hilbert space $V$, then for any continuous functional $l$, there exists $u\in V$ s.t. $B(u,v)=l(v)$.
I am wondering if this result can be extended to the case of Banach space,i.e. $B$ is defined on a Banach space $V$. By the condition of strictly positive and boundededness, we know that the topology of the Banach space is the same as the topology defined by bilinear form $B$, then we can use Riesz reprensentation theorem to prove it.