I have a homework problem involving a Hilbert space and letting $B:H\times H \rightarrow \mathbb{C}$ be a functional which is linear with respect to the first argument and conjugate linear with respect to the second argument.
I am not sure what this means, and I am trying to research it but I'm not using good keywords. I am trying to learn as much as I can as quickly as I can, so I would appreciate some direction for rigorous understanding of these terms and concepts. (I have a textbook...) I just don't understand this question.
Thank you
Linear in the first argument means that for all $x,y,z\in H$ and $\alpha\in\mathbb C$, $$B(\alpha x+y,z)=\alpha B(x,z)+B(y,z).$$ That is, for each $z\in H$, the map $x\mapsto B(x,z)$ is linear.
Conjugate linear in the second argument means that for all $x,y,z\in H$ and $\alpha\in\mathbb C$, $$B(z,\alpha x+y)=\overline\alpha B(z,x)+B(z,y).$$ That is, for each $z\in H$, the map $x\mapsto\overline{B(z,x)}$ is linear.