 Solution

For fixed $y$, Riesz gives a unique $Sy$ with $\theta (x,y)=⟨x,Sy⟩$. Conjugate linearity in $y$ and the inner-product convention make $S$ linear. Moreover $\Vert Sy\Vert ={\sup }_{\Vert x\Vert \le 1}|\theta (x,y)|\le \Vert \theta \Vert \Vert y\Vert$, and taking both suprema proves $\Vert S\Vert =\Vert \theta \Vert$.