formally

If the data set is large enough then there is no way to represent it as a functional relationship between finite dimensional vector spaces. In fact, this problem already is visible in existing large neural networks because they can only work with data that conforms to the dimensional constraints of the input space. It's why image transformers trained on NxM images don't work on any other grid size.

This is only true if the codomain has the relevant structure for vector operations. Functions are more general than vectors.

You should look into the pigeonhole principle.