Here $H$ is a Hilbert space and $H^{*}=B(H,F)$ is the dual space. I understand what is in this picture up to the last line. I have successfully shown that $||f_{y}||_{H^{*}}\leq||y||_{H}$ however I am having difficulty showing that $||y||_{H}\leq||f_{y}||_{H^{*}}$. I tried showing that $|f_{y}(y)|\leq||f_{y}||_{H^{*}}$ however had trouble with the fact that $||y||$ may be larger than 1. What is the correct approach for this?
As @guy3134 indicates, look at the definition of the norm.
In other words, observe that $$\|f_y\|_{H^*}=\sup_{\|x\|\leq 1, x\in H} |f_y(x)|\geq \bigg|f_y\left( \frac{y}{\|y\|_H} \right) \bigg|=\left<\frac{y}{\|y\|_H},y \right>=\frac{\left< y,y \right>}{\|y\|_H}=\frac{\|y\|_H^2}{\|y\|_H}=\|y\|_H$$ because the norm of $\frac{y}{\|y\|_H}\in H$ is smaller than equal to $1$.