Chapter 5 - Inner Product Spaces - 5.1 Definition of an Inner Product Space - Problems - Page 352: 35

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Assume $V$ be an inner product space. Since $v,w \in V$ we have a) $$||v+w||^2\\ =(\sqrt (v+w,v+w)^2\\ =(v,v)+(v,w)+(w,v)+(w,w)\\ =||v||^2+2(v,w)+||w||^2$$ b) Since $v,w \in V \rightarrow (v,w)=0$ then $$||v+w||^2=||v||^2+2(v,w)+||w||^2\\ =||v||^2+||w||^2$$