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.
Best Answer
As you said, functions define relations between the different "dimensions" of the space. This idea is made explicit via correlation functions in Gaussian processes (this is an excellent non-engineering introduction) which can be framed in the theory of Reproducing Hilbert Kernel Spaces (RHKS).
Based on your comment, I think you might want to look at the article,
Therein Sobolev spaces are studied in relation with their RHKS. This is an engineering article and not a mathematics one; nevertheless, it provides lots of references for further reading.