Show that an inner product space satisfies the parallelogram law with its induced norm: \[ \lVert x+y\rVert^2+\lVert x-y\rVert^2=2\lVert x\rVert^2+2\lVert y\rVert^2. \] (For simplicity, you may assume that this is an inner product space over \(\mathbb R\).)
Solution
1. Expand both sides using bilinearity and symmetry of the inner product.
For \(x, y\) in a real inner product space, \[\lVert x+y\rVert^2 = \langle x+y, x+y\rangle = \lVert x\rVert^2 + 2\langle x, y\rangle + \lVert y\rVert^2,\] \[\lVert x-y\rVert^2 = \langle x-y, x-y\rangle = \lVert x\rVert^2 - 2\langle x, y\rangle + \lVert y\rVert^2.\]
2. Add.
Adding the two identities, the cross terms cancel: \[\lVert x+y\rVert^2 + \lVert x-y\rVert^2 = 2\lVert x\rVert^2 + 2\lVert y\rVert^2.\]
3. Q.E.D.