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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.

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    @JuanPi Thanks, this is exactly what I'm looking for. Can you make that an answer rather than a comment so I can accept it? If I understand correctly, I could apply this to a Sobolev space by taking a power of the Laplacian as the reproducing Kernel, and for a graph, a power of the graph Laplacian.2015-09-26

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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,

Schaback, R., & Wendland, H. (2006). Kernel techniques: from machine learning to meshless methods. Acta Numerica, 1–97. http://doi.org/10.1017/S0962492904000077

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.