Is every symmetric bilinear form on a Hilbert space a weighted inner product?
i.e. can I write that $b(u,v) = (wu,v)_H$ for all $u, v \in H$?
I am not sure about this. Maybe something to do with Riesz theorem..
Since you’re interested in bilinear forms, I’ll assume you’re working with a real Hilbert space $H$; in the complex case, replace “bilinear” with “sesquilinear.”
Throughout all this, all you really need is to remember what the Riesz theorem says, and what it means for an operator to be self-adjoint and positive-definite. In particular, observe the role (indeed, necessity and sufficiency) of the hypotheses of boundedness, symmetry (which you already had), and positive-definiteness.