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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, Cham: Springer (ISBN 978-3-319-64610-7/hbk; 978-3-319-64612-1/ebook). xxii, 668 p. (2017). ZBL1454.81004.

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    $\begingroup$ See math.stackexchange.com/questions/226736/…. $\endgroup$
    – KConrad
    Commented Jan 2, 2022 at 4:16
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    $\begingroup$ See also mathoverflow.net/questions/20268/… $\endgroup$
    – KConrad
    Commented Jan 2, 2022 at 4:30
  • $\begingroup$ Sadly, the quote’s clarifying phrase “given as a set of points” invalidates the approach described, which depends on the structure of M. Deleting that phrase and replacing “the way” with “one good way” would lead to an accurate statement, but the actual quote is too wrong to be worth using. $\endgroup$
    – user44143
    Commented Jan 7, 2022 at 7:28

4 Answers 4

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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).

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  • $\begingroup$ I would also include Banach spaces and bounded linear functionals - that is their dual spaces… $\endgroup$
    – Pietro Majer
    Commented Jan 10, 2022 at 16:38
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The idea goes back to the 1930s when algebraic geometers understood that points of an algebraic variety "are" maximal (or prime) ideals of the ring of regular functions on it. The counterpart of this in analysis is the theory of commutative Banach algebras of Gelfand. Later Grothendieck revised the foundations of the whole algebraic geometry based on this idea, and it spread to many other areas of mathematics.

To answer the comment: van der Waerden's Moderne Algebra, 1-st edition was published in 1930. In it a point (of a Riemann surface) is defined as a certain subring of the field (of meromorphic functions on this Riemann surface). And surely, van der Waerden is not the author of this idea: his book is based on lectures of Artin and Noether.

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    $\begingroup$ 1930? Are you sure? As far as I know, the work of Marshall Stone on boolean algebras was motivated by quantum mechanics and preceded the work of Gelfand or of the algebraic geometers. en.wikipedia.org/wiki/… $\endgroup$
    – coudy
    Commented Jan 6, 2022 at 20:20
  • $\begingroup$ @coudy: I added an answer to your comment to my answer on the question. $\endgroup$
    – Alexandre Eremenko
    Commented Jan 7, 2022 at 0:21
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    $\begingroup$ @coudy Stone's paper appeared in 1936. The correspondence between points on an algebraic variety (in affine space over $\mathbf C$) and maximal ideals in a polynomial ring is based on Hilbert's Nullstellensatz (1890). Work on giving algebraic foundations to algebraic geometry based on ideal theory was done in the 1920s and 1930s (e.g., generic points by van der Waerden in 1926). This preceded Stone's work. $\endgroup$
    – KConrad
    Commented Jan 7, 2022 at 1:05
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Morse theory is an example of such method. The classification of compact surfaces using Morse theory is done for example in the book of Hirsch, Differential topology. The book of Milnor, Lectures on the h-cobordism theorem goes a step further by proving the Poincare conjecture in dimension bigger than five using Morse theory.

It should be emphasized however that "look at $F(M)$, the set of functions on $M$" is just one way to understand a space $M$ amongst many others.

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The (finite covering) dimension of $T_{3\frac12}$-topological spaces (completely regular spaces) is characterized by the universal maps into respective cubes.

(Universal maps are not just "into" but always surjective, i.e. onto -- but that's not enough)

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