Let $H_1 \times H_2$ be the product of two inner product spaces. Show that
$$<(x_1,x_2),(y_1,y_2)> = < x_1,y_1>_{H_1} + < x_2,y_2 >_{H_2}$$
defines an inner product in $H_1 \times H_2$.
I am trying to show the property that $< u+v,w > = < u,w > + < v,w >$ however not managing to get to an answer
Best Answer
Trying writing $u=(u_1,u_2) , v=(v_1,v_2) , w= (w_1,w_2)$. Then, using your definition you can proceed as: $$<u+v,w>= <(u_1,u_2)+(v_1,v_2), (w_1,w_2)> = < (u_1+v_1, u_2+v_2),(w_1,w_2)> = $$ $$ =<u_1+v_1,w_1>_{H_1} + <u_2 +v_2, w_2>_{H_2}$$
Which then takes the form $$ <u+v,w>= <u_1,w_1>_{H_1} + <v_1,w_1>_{H_1} + <u_2,w_2>_{H_2} + <v_2,w_2>_{H_2} = $$ But we can pair them together, using our definition ( $<u_1,w_1>_{H_1} with <u_2,w_2>_{H_2}$ and the other two together: $$ <u+v,w>= <(u_1,u_2),(w_1,w_2)> + <(v_1,v_2), (w_1,w_2)> = <u,w> + <v,w> $$