If you pick a random vector in $\mathbb{R}^n$ with some fixed basis, there is no special relationship between components. The relationship between the $1^{st}$ component and the $5^{th}$ component is the same as the relationship between the $82^{nd}$ component and the $1001^{th}$ component.
On the other hand, if the space $\mathbb{R}^n$ is viewed as a discretization of a function space (eg, n nodal values for a piecewise linear basis of hat functions), then there is a special relationship between components based on nearness in the underlying domain. If 2 nodes are close in physical space, then the basis vectors corresponding to those nodes are more highly related in the function space.
So, somehow $\mathbb{R}^n$ as a function space has more structure and is different than $\mathbb{R}^n$ generically. What is this difference and how can it be made precise?
My thoughts so far are as follows: this seems similar to the ideas of function space regularity (the more regular the space, the more nearby points are "related" to each other). However I don't think this is the whole picture since one could also imagine defining additional structure on the function space over nodes in a n-node graph $\{f:G\rightarrow\mathbb{R}\}$, where there is no notion of continuity, differentiability, etc.