Functional in Hilbert space Let $H$ be a Hilbert space and $0\neq x\in H$. I want to prove that there is an unique $f\in H^*$, such that $\|f\|=1$ and $f(x)=\|x\|$.
Any ideas on how to approach this problem

Call such functional $f$ norming. Let us try to convince us that Riesz's theorem is not quite necessary here.
Let $X$ be a Banach space and $x\in X$ be a non-zero vector. Then $x$ has a unique norming functional whenever the function $t\mapsto \|x+ty\|$ is differentiable at 0 for every $y\in X$. Please see here for more hints. In our case (that is for Hilbert spaces), this function is certainly differentiable at 0 given the shape of the unit sphere of the two-dimensional Euclidean space.
Here $f$ can be found explicitly—take $f=\frac{1}{\|x\|}x$.
