Result in dual Hilbert space

Question

I study about bounded linear functional on Hibert space and I came up to a result that I can't understand how we took it.

We have φ a bounded linear functional and it says that If $y\in H$ then, $φ_{y}(x)=<y,x>$ ( which we take it from Riesz representation I think) is a bounded linear functional on H with , $\left \| φ_{y} \right \|=\left \| y \right \|$.

And my question is : why the last equation holds ??

Thank you in advance.

Answer 1

$|\phi_y(x)|\leq \|x\|\|y\|$ Cauchy-Schwarz

and $|\phi_y(y/\|y\|)|=|\langle y,y/\|y\|\rangle|=\|y\|$.