Transport of an Inner Product and of Hilbert Space Structure along a Linear Bijection

In the setting of Real Hilbert Spaces: Standing Notation and Background, whose standing space $H$ is not used here, let $X$ be a vector space over $\mathbb{R}$ with zero vector $0_{X}$, let $E$ be a real inner product space with inner product $\langle\cdot,\cdot\rangle_{E}$, norm $|\cdot|_{E}$, distance $d_{E}$ and zero vector $0_{E}$, let $T:X\to E$ be a linear bijection, and let $T^{-1}:E\to X$ be its inverse. For $x,y\in X$ put

$$\langle x,y\rangle_{X}=\langle Tx,Ty\rangle_{E}.$$

Then the following hold.

1. (The transported inner product)(i) $\langle\cdot,\cdot\rangle_{X}$ is an inner product on $X$, so that $X$ with it is a real inner product space; writing $|\cdot|_{X}$ and $d_{X}$ for its norm and distance, $|x|_{X}=|Tx|_{E}$ and $d_{X}(x,y)=d_{E}(Tx,Ty)$ for all $x,y\in X$.

2. (The inverse)(ii) $T^{-1}$ is linear, and $\langle T^{-1}a,T^{-1}b\rangle_{X}=\langle a,b\rangle_{E}$ for all $a,b\in E$.

3. (Completeness)(iii) If $E$ is a real Hilbert space, then $X$ with $\langle\cdot,\cdot\rangle_{X}$ is a real Hilbert space.

4. (Orthonormal bases)(iv) Suppose $E$ is a real Hilbert space and $(e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $E$. Then $(T^{-1}e_{k})_{k\in\mathbb{N}}$ is an orthonormal basis of $X$.