I was dealing with this:
Let
\begin{equation} [ \cdot, \cdot ] : H \times H \rightarrow \mathbb{C} \end{equation}
such that $\forall x,y,z \in H$, $\lambda, \mu \in \mathbb{C}$, $$ \langle x, \lambda y+\mu z\rangle = \lambda \langle x, y \rangle + \mu \langle x, z \rangle,$$ $$\langle \lambda y+ \mu z, x \rangle = \bar{\lambda} \langle y,x \rangle + \bar{\mu} \langle z,x \rangle$$ $$\exists c\ge 0 : |\langle x,y \rangle| \le c\|x\| \|y\|.$$
Show that there exists $A \in B(H)$ such that \begin{equation} [ x,y ] = \langle Ax,y \rangle \end{equation} (Bounded sesquilinear form) and \begin{equation} \|A\| = \min\{c \ge 0 : |\langle x, y \rangle| \le c\|x\| \|y\| \} \end{equation}
Of course one needs to recall the Riesz representation theorem to do this, but how, formally?
I will use the common convention in mathematics (as opposed to physics) where the conjugate linear coordinate is the second one.
For each $y$, consider the map $\phi_y(x)=x\longmapsto [x,y]$. This is linear, and $$ |\phi_y(x)|=[x,y]|\leq c \|x\|\,\|y\|. $$ So $\|\phi_y\|\leq c\|y\|$. By the Riesz Representation Theorem, there exists $z_y$ such that $[x,y]=\langle x,z_y\rangle$. So we have a function $B:H\to H$, given by $By=z_y$. This $B$ is linear by the uniqueness in the RRT. Also, $$ \|By\|=\sup\{|\langle x,By\rangle|:\ \|x\|=1\}=\sup\{|[x,y]|:\ \|x\|=1\}\leq c\|y\|. $$ So $B$ is bounded with $\|B\|\leq c$, and if $A=B^*$, then $\|A\|=\|B^*\|=\|B\|\leq c$, and $$ [x,y]=\langle x,By\rangle=\langle Ax,y\rangle. $$