I have a question concerning the precise handling the usual function spaces like $L^2$ in the context of the superstructure. In their paper
Benci, Vieri; Luperi Baglini, Lorenzo. Generalized solutions in PDEs and the Burgers' equation. J. Differential Equations 263 (2017), no. 10, 6916-6952 (DOI: 10.1016/j.jde.2017.07.034, arXiv:arXiv:1602.01593).
the authors work with a suitable space $V(\Omega)$ of real-valued functions defined in a Euclidean domain $\Omega$. In order to be able to develop generalized solutions to Burgers' equation, the authors construct an extension $V_\Lambda(\Omega)$ consisting of certain internal hyperreal-valued functions. The construction involves a fairly large index set $\mathcal P_{fin}(\mathbb V_{\infty})$ (of cardinality $\beth_\omega$) relying on the detailed structure of the superstructure. It also relies upon the precise position of $V$ with regard to the superstructure. Such placement is not something one usually worries about in functional analysis where one relies on standard identifications, but here it seems significant.
At the next stage, on page 6922, the authors specify the requirement $V\subseteq L^2$. Since $L^2$ is not a space of functions but rather a space of equivalence classes of functions, such an embedding is apparently not meant literally, and at this stage one can assume that this simply means that every function in $V$ is assumed square-integrable (alternatively, one can embed $L^2$ into the same level of the superstructure as $V$ by means of a suitable Hamel basis, but this ultimately runs into difficuties because of further identifications being employed).
On page 6923, the authors assume that $V_\Lambda\subseteq {}^\ast\!(L^2)$, which suggests that the whole space $L^2$ is embedded into the superstructure at the appropriate level so as to be able to speak of its nonstandard extension (they actually use the notation with the asterisk on the right side, which I will avoid here).
On page 6924, one finds the inclusion $L^2 \subseteq V'$ (topological dual of $V$, i.e., set of distibutions). This again raises the question of how these spaces are embedded in the superstructure exactly.
By page 6925, the authors are talking about projections ${}^\ast\! L^2\to V_\Lambda$, as well as intersections $L^1_{loc}\cap L^2$, etc. Such intersections of course also rely on certain standard identifications commonly used in functional analysis, but in the context of the construction of $V_\Lambda$ which involves the details of the location of $V$ in the superstructure it is not clear to me how to handle such identifications.
Question. How are the issues of the precise superstructure embedding of standard function spaces handled in the context of nonstandard extensions?
