On the compactness of families of continuous functions in Grand Lebesgue spaces

In this paper, we established compactness results for families of continuous functions in grand Lebesgue spaces $G_{\psi }$, where $\psi :(a,b)\to (0,\mathrm{\infty })$ is a given positive function that determines the structure of the space. In particular, we showed that equicontinuity together with uniform boundedness implies relative compactness in the $G_{\psi }$-norm. The approach is based on Arzelà–Ascoli-type arguments combined with uniform control in grand Lebesgue space (GLS) norms. Several examples were provided to illustrate convergence mechanisms, and extensions to weighted GLS and fractional Sobolev-type settings are discussed. These results establish a unified compactness framework in grand Lebesgue spaces and provide a foundation for further investigations in Sobolev-type embeddings and related analytical models.