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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.

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What do you want to know? Certainly people think about such things, e.g. these are the forms that the complex orthogonal groups $\text{O}(n, \mathbb{C})$ preserve. –  Qiaochu Yuan Nov 5 '13 at 23:56
It is hard to specify exactly. I just have a feeling that the right reference will contain some missing pieces of information. For example, are there representation theorems available for complex bilinear forms akin the "standard" result for Hermitian sesquilinear below bounded forms? Moreover, the concept of complex differentiability seems to me should be central. –  Simen K. Nov 7 '13 at 8:58

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