The Product of Two Real Inner Product Spaces

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $E_{1}$ and $E_{2}$ be real inner product spaces, with inner products $\langle\cdot,\cdot\rangle_{E_{1}}$ and $\langle\cdot,\cdot\rangle_{E_{2}}$ and zero vectors $0_{E_{1}}$ and $0_{E_{2}}$, and let $E_{1}\times E_{2}$ be the Cartesian product of the underlying sets, whose elements are the pairs $(x_{1},x_{2})$ with $x_{1}\in E_{1}$ and $x_{2}\in E_{2}$, two such pairs being equal exactly when their corresponding components are equal.

1. (The product)(i) The product of $E_{1}$ and $E_{2}$ is the set $E_{1}\times E_{2}$ equipped with the addition and scalar multiplication given by

$$(x_{1},x_{2})+(y_{1},y_{2})=(x_{1}+y_{1},\,x_{2}+y_{2}), \qquad \lambda\,(x_{1},x_{2})=(\lambda x_{1},\,\lambda x_{2})\quad\text{for }\lambda\in\mathbb{R},$$

and with the product pairing, the map assigning to each pair of elements $(x_{1},x_{2})$ and $(y_{1},y_{2})$ of $E_{1}\times E_{2}$ the real number

$$\langle(x_{1},x_{2}),(y_{1},y_{2})\rangle=\langle x_{1},y_{1}\rangle_{E_{1}}+\langle x_{2},y_{2}\rangle_{E_{2}} .$$

2. (Coordinate maps)(ii) The coordinate maps of the product are the maps $\pi_{1}:E_{1}\times E_{2}\to E_{1}$ and $\pi_{2}:E_{1}\times E_{2}\to E_{2}$ given by $\pi_{1}(x_{1},x_{2})=x_{1}$ and $\pi_{2}(x_{1},x_{2})=x_{2}$, and its coordinate injections are the maps $j_{1}:E_{1}\to E_{1}\times E_{2}$ and $j_{2}:E_{2}\to E_{1}\times E_{2}$ given by $j_{1}x=(x,0_{E_{2}})$ and $j_{2}x=(0_{E_{1}},x)$.