Let $\mathcal{H}$ be a Hilbert space and $B: \mathcal{H} \times \mathcal{H} \rightarrow \mathbb{C}$ be a functional which is linear with respect to the first argument and conjugate linear with respect to the second argument. Assume, furthermore that there exists an $M > 0$ such that $|B(f,g)| \leq M||f||||g||$ for all $f,g \in \mathcal{H}$. Prove that there exists a unique linear operator $T: \mathcal{H} \rightarrow \mathcal{H}$ such that $B(f,g)=(Tf,g)$ for all $f,g \in \mathcal{H}$. Show also that one can take $M=||T||$.
...I firstly need to understand the context of this question better...I am supposing "linear with respect to the first argument and conjugate linear with respect to the second argument" means that the function is linear in $H \times H$, and it is conjugate linear in $\mathbb{C}$. (Is this what that means?) Then, since $M$ exists there is a maximum value...but how can we use this to find a unique linear operator?