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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'?

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Let $S=\{x\in H:\langle x,a\rangle,\langle x,b\rangle\ge 1\}$, the set of all $x\in H$ satisfying the desired condition, and let $N=\{\|x\|:x\in S\}$. You have to do two things:

  1. Show that $N$ has a smallest member. In other words, show that $\inf N=\min N$. Call that number $m$.
  2. Show that there is exactly one $x\in S$ such that $\|x\|=m$.

In other words, out of all $x$’s satisfying the condition, there is exactly one that has the smallest norm.

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