\CySpT^{m, \, 3}:~ \eta_{m, \, 3}(u)
    \in \CaSpB^2}~. 
\end{aligned}$$ Then, since proper conforming discretizations are known for the spaces in Cartesian coordinates –, the discrete spaces $\CySpT_h^{m, \, k}$ are built from the Cartesian discrete spaces using $\eta_{m, \, k}^{-1}$, *i.e.* $$\label{eq:cyl_phys_disc_sps}
\begin{aligned}
\CySpT_h^{m, \, 0} & = \parG{u_h ~|~ u_h = \eta_{m,
        \,0}^{-1}\parT{\ut_h}, ~ \ut_h \in \CaSpB_h^0}~,\\ 
\CySpT_h^{m, \, 1} & = \parG{\uv_h ~|~ \uv_h = \eta_{m, \,
        1}^{-1}\parT{\uvt_h}, ~ \uvt_h \in \CaSpB_h^1 \times
    \CaSpB_h^0}~,\\ 
\CySpT_h^{m, \, 2} & = \parG{\uv_h ~|~ \uv_h = \eta_{m, \,
        2}^{-1}\parT{\uvt_h}, ~ \uvt_h \in \CaSpB_h^{1*} \times
    \CaSpB_h^2}~,\\ 
\CySpT_h^{m, \, 3} & =\parG{u_h ~|~ u_h = \eta_{m, \,
        3}^{-1}\parT{\ut_h}, ~ \ut_h \in \CaSpB_h^2}~. 
\end{aligned}$$ Consistently with this definition, we denote functions defined on spaces in Cartesian coordinates with a tilde. Moreover, since in the following discussion the $\rho$ and $z$ components will be often treated differently from the $\theta$ component, we will denote with the subscript $\rho z$ the meridian component of a function $\uv_{\rho z} = \parTs{u_\rho, \,
    u_z}^T$. In order to apply the differential operators to the discrete functions, it is useful to define, by composition, the following operators that act directly on the continuous spaces defined in Cartesian coordinates: $$\begin{aligned}
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