Inequality proof (Hilbert space)

Question

Show that if H is a Hilbert space, then: $$\Vert(x+y)\Vert^2 - \Vert(x - y)\Vert^2 \le 4 \Vert x\Vert \Vert y\Vert, $$ for all $x, y \in H. $

Answer 1

Observe that $$ \Vert x\pm y\Vert^2 = \langle x\pm y,x\pm y \rangle = \Vert x \Vert^2 \pm 2\Re \langle x, y \rangle + \Vert y\Vert^2, $$ so that $$ \Vert x + y\Vert^2 - \Vert x - y\Vert^2 = 4 \Re\langle x, y \rangle. $$ Now apply Cauchy-Schwarz.