Isometry in Riesz's representation theorem Riesz's representation theorem states that if $H$ is a Hilbert space and $H^*$ its dual space, then the map $\Phi$ which maps $x\in H$ to $x^*\in H^*, x^*y:=\langle x, y\rangle \,\forall\,y\in H$ is an antilinear isometric isomorphism. The fact that it is isometric is supposed to be trivial, and is presented as "manifestly true" in all proofs I could find, but the only thing I can tell is that by the Cauchy Schwartz inequality $$||\Phi(x)||=\sup\left\\{|\langle x, y\rangle|, ||y||\leq 1\right\\}\leq\sup\left\\{||x||\,||y||, ||y||\leq 1\right\\}=||x|| $$ But why do we have equality?

We have $\|\Phi(x)\|\geq\left|\left\right|=\dfrac{\|x\|^{2}}{\|x\|}=\|x\|$ for nonzero $x$.