Chapter 5 - Inner Product Spaces - 5.1 Definition of an Inner Product Space - Problems - Page 352: 36

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Assume $V$ be an inner product space. From exercise 35, we have: $$||v+w||^2=||v||^2+2(v,w)+||w||^2$$ Thus: $||v-w||^2=||v||^2-2(v,w)+||w||^2$ then $$||v+w||^2-||v-w||^2=||v||^2+2(v,w)+||w||^2-(||v||^2-2(v,w)+||w||^2))\\ =4(v,w)$$ $$||v+w||^2+||v-w||^2=||v||^2+2(v,w)+||w||^2+(||v||^2-2(v,w)+||w||^2))\\ =2(||v||^2+||w||^2)$$