[เรียวคุง]

Let $\{ H_{1},<\cdot,\cdot>_{1}\}$ and $\{ H_{2},<\cdot,\cdot>_{2}\}$ be Hilbert Spaces.
Prove that the set of all pair $(x,y)$ with $x\in H_{1}$ and $y\in H_{2}$ is a Hilbert Space with inner product given by $$ <(x_{1},y_{1}),(x_{2},y_{2})>~=~<x_{1},x_{2}>_{1} + <y_{1},y_{2}>_{2}$$