In Quantum theory, groups and representations, Peter Woit writes:



> 
> A fundamental principle of modern mathematics is that the way to
> understand a space $M$, given as some set of points, is to look at $F(M)$,
> the set of functions on this space.
> 
> 
> 


I was wondering what some examples of this "fundamental principle" is across different fields in mathematics.


*Woit, Peter*, [**Quantum theory, groups and representations. An introduction**](https://dx.doi.org/10.1007/978-3-319-64612-1), Cham: Springer (ISBN 978-3-319-64610-7/hbk; 978-3-319-64612-1/ebook). xxii, 668 p. (2017). [ZBL1454.81004](https://zbmath.org/?q=an:1454.81004).



What are some examples of understanding a space by studying the functions on this space?
The idea of studying the relationship between structured spaces and appropriate spaces of functions thereon could be described as one of the basic principles of functional analysis, perhaps even the defining one.


Examples:


* completely regular spaces and continuous functions—general, bounded or of compact support (in the locally compact case);
* $\sigma$-algebras and (bounded) measurable functions;
* measure spaces and $L^p$-spaces (strictly speaking, equivalence classes of functions);
* smooth manifolds, including open subsets of euclidean space, and spaces of smooth functions, sometimes combined with growth conditions;


and finally, but the list could go on,


* complex manifolds and holomorphic functions, again often combined with growth conditions.


The next link in the chain is a consideration of the duals of these function spaces. Here there are two main streams:


* representation theorems—the cases where these duals have explicit descriptions, either as spaces of functions themselves or of measures (duality for $L^p$-spaces, Riesz representation theorem);
* the cases where they are used to define new types of objects (Schwartzian distributions, the Bourbakian approach to measure theory).


