What's the proper notation to indicate that a function is restricted to a particular space in only some of its arguments?
For example, in non-relativistic quantum mechanics for one particle in the position space, the wave function is written $\psi(t,\vec{x})$ and solves $i\hbar\partial_t\psi(t,\vec{x})=\hat{H}\psi(t,\vec{x})$.
This function is $\psi:\mathbb{R}\times\mathbb{R}^3\to\mathbb{C}$, but it has additional restrictions. For any constant $t$, $\psi(\vec{x})$ for $\mathbb{R}^3\to\mathbb{C}$ lies in $L^2(\mathbb{R^3})$ (with $L^2(X)$ on measure space $X$). How can this restriction be shown when expressing the domain & codomain in arrow notation?
Possibly $\psi:\mathbb{R}\to L^2(\mathbb{R}^3\to\mathbb{C})$ or $\psi:\mathbb{R}\to L^2_\mathbb{C}(\mathbb{R}^3)$? But I've never seen any notation that indicates whether $^2(Σ)$ is over $\mathbb{R}$ or $\mathbb{C}$, and different answers disagree about which is intended unless specified. How can this be notated?
(Ignore other properties of this example, e.g. $L^2$ equivalence class, projective space, C* details).