A Hahn-Banach theorem for quadratic forms (Q2732521)

Let \(E\) be a normed vector space over the scalar field \(K\left( ={\mathbb{R}}\vee{\mathbb{C}}\right) \) and let \(H\) be a hyperplane in \(E\). Then for every continuous quadratic form \(Q\) on \(H\) there exists a continuous quadratic form \(\widetilde{Q}\) on \(E\) such that \(\widetilde{Q}|_{H}=Q\) and \(\left\|\widetilde{Q}\right\|\leq C\left( K\right) \cdot\left\|Q\right\|\), where \(C\left( K\right) \) is a constant depending merely on \(K\). The author proves that \(C\left( {\mathbb{R}}\right) =2\) and \(\frac{7}{3}\leq C\left( {\mathbb{C}}\right) \leq 2\sqrt{2}.\)