Show that 12v w2 v w2 v2 w2 for any v and w in an inner

Show that 1/2(||v + w||^2 + ||v - w||^2) = ||v||^2 + ||w||^2 for any v and w in an inner product space.

Solution

Let v = (v₁ , v₂, …, v_n ,…..) and w = ( w₁ , w₂ , …, w_n , ….). Then v +w = ( v₁ +w₁ , v₂ +w₂ , …, v_n +w_n ,…) and v –w = ( v₁ -w₁ , v₂ -w₂ , …, v_n -w_n ,…) . Further, || v+w||² = [( v₁ +w₁)² + (v₂ +w₂)² +…+(v_n+w_n)² +….] = [ v₁² + w₁² + 2v₁ w₁ + v₂² + w₂² + 2v₂ w₂ + … + v_n² + w_n² + 2v_nw_n +…] and || v -w||² = [( v₁ -w₁)² + (v₂ -w₂)² +…+(v_n- w_n)² +….] =   [ v₁² + w₁² - 2v₁ w₁ + v₂² + w₂² - 2v₂ w₂ + … + v_n² + w_n² - 2v_nw_n +…] so that 1 /2[|| v+w||² + || v-w||² ] = v₁² + w₁² + v₂² + w₂² +… + v_n² +w_n² +…. = (v₁² + v₂² +…+v_n² +…) + (w₁² + w₂² + …+w_n² +…) = || v||² + ||w||²