Past exam of the analysis course of the University of Oxford / 2011 / 5 / a / ii / Solution

Put $w=x-y_0$. For any $y\in Y$ and any real $t$, minimality givesThe derivative at zero must vanish, so $\mathrm{Re}⟨w,y⟩=0$. In a complex Hilbert space, apply the same argument to $iy$ to get the imaginary part zero as well. In the real case the first conclusion is already enough. Hence