First of all, here is the assignment:
Let $X$ be a Hilbert space over $\mathbb{C}$, $V \subseteq X$ be a closed subspace and $f \in L(V, \mathbb{C}) $ a linear continuous operator. Show that there exists one unique continuation $F$ of $f$ on $X$, such that all these properties are satisfied:
- $F \in L(X, \mathbb{C})$ (i.e., linear and continuous),
- $F|_V = f$,
- $\|F\| = \|f\|$.
There is also a hint given: Use the Riesz representation theorem. Ok, so I should know where to start, but fact is: I have no real clue. I know that using Riesz theorem property 3 can easily be shown from property 2. But how do I prove that there exists this continuation? I'd be glad if anyone could give me a hint on how to approach this.