Properties of 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 norms $|\cdot|_{E_{1}}$, $|\cdot|_{E_{2}}$ and distances $d_{E_{1}}$, $d_{E_{2}}$, and let $E_{1}\times E_{2}$ be their product, with the product pairing $\langle\cdot,\cdot\rangle$ defined there and with the coordinate maps $\pi_{1},\pi_{2}$ and coordinate injections $j_{1},j_{2}$. Let $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ be elements of $E_{1}\times E_{2}$ and let $\lambda\in\mathbb{R}$. Then the following hold.

1. (The product is a real inner product space)(i) The addition and scalar multiplication of The Product of Two Real Inner Product Spaces (i) make $E_{1}\times E_{2}$ a vector space over $\mathbb{R}$, whose zero vector is $0_{E_{1}\times E_{2}}=(0_{E_{1}},0_{E_{2}})$ and in which the additive inverse of $(x_{1},x_{2})$ is $(-x_{1},-x_{2})$, and the product pairing is an inner product on that vector space. In the remaining claims $E_{1}\times E_{2}$ denotes the resulting real inner product space, whose norm and distance are written $|\cdot|$ and $d$.

2. (Norm of a pair)(ii)

$$|x|^{2}=|x_{1}|_{E_{1}}^{2}+|x_{2}|_{E_{2}}^{2}, \qquad |x_{1}|_{E_{1}}\le|x|,\quad|x_{2}|_{E_{2}}\le|x|,\quad |x|\le|x_{1}|_{E_{1}}+|x_{2}|_{E_{2}} .$$

3. (Comparison with the product metric)(iii) One has $d(x,y)^{2}=d_{E_{1}}(x_{1},y_{1})^{2}+d_{E_{2}}(x_{2},y_{2})^{2}$. Writing $\rho$ for the product metric of $(E_{1},d_{E_{1}})$ and $(E_{2},d_{E_{2}})$, one has

$$\rho(x,y)\le d(x,y)\le 2\,\rho(x,y),$$

and consequently the metric spaces $(E_{1}\times E_{2},d)$ and $(E_{1}\times E_{2},\rho)$ have the same convergent sequences with the same limits, the same Cauchy sequences, and the same open and closed subsets.

4. (Componentwise convergence)(iv) Let $(z_{m})_{m\in\mathbb{N}}$ be a sequence in $E_{1}\times E_{2}$ and let $z\in E_{1}\times E_{2}$. Then $(z_{m})$ converges to $z$ in $(E_{1}\times E_{2},d)$ if and only if $(\pi_{1}z_{m})_{m\in\mathbb{N}}$ converges to $\pi_{1}z$ in $(E_{1},d_{E_{1}})$ and $(\pi_{2}z_{m})_{m\in\mathbb{N}}$ converges to $\pi_{2}z$ in $(E_{2},d_{E_{2}})$; and $(z_{m})$ is a Cauchy sequence in $(E_{1}\times E_{2},d)$ if and only if $(\pi_{1}z_{m})$ and $(\pi_{2}z_{m})$ are Cauchy sequences in $(E_{1},d_{E_{1}})$ and $(E_{2},d_{E_{2}})$ respectively.

5. (Coordinate maps and injections)(v) The maps $\pi_{1}$ and $\pi_{2}$ are linear and satisfy $|\pi_{i}x|_{E_{i}}\le|x|$ for $i\in[2]$, so that $\pi_{1}\in\mathcal{L}(E_{1}\times E_{2},E_{1})$ and $\pi_{2}\in\mathcal{L}(E_{1}\times E_{2},E_{2})$. The maps $j_{1}$ and $j_{2}$ are linear and satisfy $|j_{1}u|=|u|_{E_{1}}$ for $u\in E_{1}$ and $|j_{2}v|=|v|_{E_{2}}$ for $v\in E_{2}$, so that $j_{1}\in\mathcal{L}(E_{1},E_{1}\times E_{2})$ and $j_{2}\in\mathcal{L}(E_{2},E_{1}\times E_{2})$. Moreover $x=j_{1}\pi_{1}x+j_{2}\pi_{2}x$.

6. (Completeness)(vi) If $E_{1}$ and $E_{2}$ are real Hilbert spaces, then $E_{1}\times E_{2}$ is a real Hilbert space.

7. (Separability)(vii) If $(E_{1},d_{E_{1}})$ and $(E_{2},d_{E_{2}})$ are separable, then $(E_{1}\times E_{2},d)$ is separable.