On page 124 of Thaller's The Dirac equation the following space is mentioned : $C^{\infty}_0(0, \infty)\otimes C^\infty(\mathbb{S}^2)\subset L^2(0, \infty)\otimes L^2(\mathbb{S}^2),$ where the symbol $\otimes$ refers to Hilbert space tensor product (cfr. Reed & Simon vol.I section II.4). Precisely, the author claims that $C_0^{\infty}(0, \infty)\otimes C^\infty(\mathbb{S}^2)$ is the image of $C^\infty_0(\mathbb{R}^3\setminus\{o\})$ under the unitary mapping
\begin{array} \.L^2(\mathbb{R}^3) &\to& L^2(0, \infty)\otimes L^2(\mathbb{S}^2) \\ \psi(x, y,z)&\mapsto& r \psi(r\sin \theta\cos\phi, r \sin \theta\sin \phi, r\cos\theta). \tag{U}\end{array}
Problem
The construction of Hilbert space tensor product involves taking a completion and so, if I'm not mistaken, it must be
$C_0^{\infty}(0, \infty)\otimes C^\infty(\mathbb{S}^2)=L^2(0, \infty)\otimes L^2(\mathbb{S}^2). $
So something is wrong. Am I mistaken? If not, what is the correct image of $C^\infty_0(\mathbb{R}^3\setminus\{o\})$ under the mapping (U)?
Thank you.