I'm reading a paper that has referred to functions in and bases for both $\mathfrak{L}^2(\mathbb{R})$ and $\mathcal{L}^2(\mathbb{R})$.
I'm familiar with the latter notation which consists of functions whose absolute value squared have finite Lebesgue integral. What does the $\mathfrak{L}$ notation mean? The paper uses it in the context of a vector-valued function and tensor products but provides no other context. Could anyone point me in the right direction as to the name of this set so that I can look further into it?
This is the paper and the notation is seen on page 2807 if that is helpful.
Edit: Here's the context in which each appears,
...and the response $Z\in[0,1]$... start by specifying an orthonormal basis $(\phi_i)_{i\in\mathbb{N}}$ for $\mathcal{L}^2(\mathbb{R})$....
For instance, if $\mathbf{Z} \in \mathbb{R}^2$, consider the basis $\{\phi_{i,j}(\mathbf{z})=\phi_i(z_1)\phi_j(z_2)\,:\,i,j\in\mathbb{N}\}$, where $\mathbf{z} = (z_1, z_2)$, and $\{\phi_i(z_1)\}_i$ and $\{\phi_j(z_2)\}_j$ are bases for functions in $\mathfrak{L}^2(\mathbb{R})$.