I do not seek a proof of the following exercise. I just want to understand this question in order to solve it myself.
Let $H$ be a Hilbert space over $\mathbb R$ and let $a, b\in H$ be such that $\langle a, b\rangle> 0$. I want to prove that there exists a unique element $x\in H$ of minimal norm such that $\langle x,a\rangle, \langle x,b\rangle\ge 1$.
But what is meant by `element $x\in H$ of minimal norm'?