I am pretty much stucked at this exercise. I have seen a couple of examples of it, but now I need to show that this is true in general.
I know it suffices to find some element $y \in H$ such that its distance with $H_0$ is not attained. To define a closed subspace in such an abstract space I have considered taking the kernel of some continuous linear functional, maybe the inner product itself.
Now, we should definetively take some Cauchy sequence $\{ x_i \}$ that converges to some $x$ which is not in $H$ and use it maybe to show that the distance of some $y$ with $H_0$ is not attained.
Could someone please help me out? Thanks!