Let $(H,(\cdot,\cdot))$ be a pre-Hilbert space on $\mathbb{K}$. A bilinear form $B:H\times H\rightarrow\mathbb{K}$ is:
bounded if there is a $C>0$ so that, $$|B(x,y)|\le C||x|| \ ||y|| \quad \forall x,y\in H$$
continuous, if for $x_n \rightarrow x$ and $y_n\rightarrow y$ the following is true: $$B(x_n,y_n)\rightarrow B(x,y)$$ for $n\rightarrow \infty$
i) A Bilinear form is bounded if and only of it is continuous
ii) Is $A:H\rightarrow H$ linear then B(x,y):=(Ax,y) is a Bilinear form B on H. B is continuous if and only if A is continuous. In this case $||A||=\sup_{x,y\neq 0}\frac{|B(x,y)|}{||x||||y||}$
I already proved i) and why $B(x,y):=(Ax,y)$ is a bilinear form. But I don't know how to show the rest if ii). The statement seems obvious but I don't know how to calculate the norm of A. The only theorem I know, which helps to calculate the norm, defines the norm as: $||A||:=\sup_{||x||\le1}||Tx||_Y<\infty$. Can someone help me?
Hints: