Let $H$ be a Hilbert space. Show that $M\subset H$ is bounded if and only if $\sup\{|\langle x,y \rangle|:y\in M\}<\infty$ for every $x\in H$
My attempt:
Since $H$ is a Hilbert space any set is bounded if and only if it is weak bounded, hence
$$\forall\lambda\in X'\exists c_{\lambda}>0:\sup_{x\in M}|\lambda(x)|\leq c_{\lambda} $$
For each lambda we have a $x_T$ such that $\lambda(x)=\langle x,x_T \rangle$ for all $x\in H$. Since $\lambda$ is bounded on $M$ this does also hold for $\langle x,x_T \rangle$.
Just an idea. Can someone look at it and improve it? The hint also tells that we should use the uniform boundedness principle but in which step?
Thanks in advance.