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

Let $d={\inf }_{y\in Y}\Vert x-y\Vert$ and choose $y_n\in Y$ with $\Vert x-y_n\Vert \to d$. The parallelogram law givesThe right side tends to zero. Thus $(y_n)$ is Cauchy. The closed subspace $Y$ of a Hilbert space is complete, so $y_n\to y_0\in Y$. Continuity of the norm gives $\Vert x-y_0\Vert =d$, proving the distance is attained.