I am working on a problem involving bilinear forms over complex Hilbert spaces, and in my case it is *not* natural to make the forms sesquilinear, i.e., $a(u,v)$ is linear in both complex arguments. Moreover, it is natural for me to consider not the usual inner product, but the bilinear scalar product, i.e., $(u,v) = \langle \bar{u},v\rangle$.

I assume there exists some standard reference, working with this setting instead of the usual assumptions of sesquilinear Hermitian forms. If someone has any idea, I would be grateful.