Banach-valued graph limits: Graphon representability and Banach-space structure

We study a graph-limit problem for Banach-decorated graphs. Given a sequence of $X$-decorated graphs whose homomorphism densities converge against all $X^*$-decorated test graphs, we ask whether the limiting densities are represented by an $X$-valued graphon. The results connect this graph-limit problem with Banach-space structure. If $X^*$ is separable, then the graphon representation property for graph sequences uniformly bounded in $L^p$ for every finite $p$ holds if and only if $X$ is reflexive. For Banach lattices, it is equivalent to the Radon--Nikod\'ym property. For dual Banach spaces, it is equivalent to the conjunction of the Radon--Nikod\'ym property and weak sequential completeness. In the bounded setting, the same characterization extends to arbitrary Banach spaces: for every Banach space $X$, the representation property for uniformly $L^\infty$-bounded graph sequences holds if and only if $X$ has the Radon--Nikod\'ym property and is weakly sequentially complete.