Let \(H\) be a Hilbert space and let \(u\) and \(v\) be two distinct elements of \(H\), both of norm \(1\).
Show that \(\frac {u+v}{2}\) has norm strictly less than \(1\). Conclude that the unit sphere \(S=\{w\in H:\lVert w\rVert =1\}\) cannot contain a nontrivial interval (i.e. an interval with distinct endpoints).