An important question inmathematics is whether a space can be completely described by the functions defined on it—that is, by its "observables." TheBanach–Stone theorem is a classical result in this direction. It shows that certain well-behaved spaces (specifically,compactHausdorff spaces) can be recovered from theBanach space ofcontinuous functions defined on them. The theorem is named after themathematiciansStefan Banach andMarshall Stone.

In brief, the Banach–Stone theorem allows one to recover acompact Hausdorff spaceX from the Banach space structure of the spaceC(X) of continuous real- or complex-valued functions onX. If one is allowed to invoke the algebra structure ofC(X) then recoveringX is easy – we can identifyX with thespectrum ofC(X), the set ofalgebra homomorphisms into the field of scalars, equipped with the weak*-topology inherited from thedual spaceC(X)*. What makes the Banach–Stone theorem striking is that it avoids reference to multiplicative structure by recoveringX from the extreme points of the unit ball ofC(X)*.

Thus the Banach–Stone theorem states that ifC(X) andC(Y) are isometrically isomorphic as Banach spaces, thenX andY are homeomorphic.

For acompactHausdorff spaceX, letC(X) denote theBanach space of continuous real- or complex-valuedfunctions onX, equipped with thesupremum norm ‖·‖_∞.

Given compact Hausdorff spacesX andY, supposeT : C(X) → C(Y) is asurjectivelinear isometry. Then there exists ahomeomorphismφ : Y → X and a functiong ∈ C(Y) with

${\displaystyle |g(y)|=1\text{ for all }y\in Y}$

such that

${\displaystyle (Tf)(y)=g(y)f(\phi (y))\text{ for all }y\in Y,f\in C(X).}$

The case whereX andY are compactmetric spaces is due to Banach,^[1] while the extension to compact Hausdorff spaces is due to Stone.^[2] In fact, they both prove a slight generalization—they do not assume thatT is linear, only that it is anisometry in the sense of metric spaces, and use theMazur–Ulam theorem to show thatT is affine, and so${\displaystyle T-T(0)}$ is a linear isometry.

The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, ifE is aBanach space with trivialcentralizer andX andY are compact, then every linear isometry ofC(X; E) ontoC(Y; E) is astrong Banach–Stone map.

A similar technique has also been used to recover a spaceX from the extreme points of the duals of some other spaces of functions onX.

The noncommutative analog of the Banach-Stone theorem is the folklore theorem that two unital C*-algebras are isomorphic if and only if they are completely isometric (i.e., isometric at all matrix levels). Mere isometry is not enough, as shown by the existence of a C*-algebra that is not isomorphic to its opposite algebra (which trivially has the same Banach space structure).

• Banach space – Normed vector space that is complete

• Araujo, Jesús (2006). "The noncompact Banach–Stone theorem".Journal of Operator Theory.55 (2):285–294.ISSN 0379-4024.MR 2242851.
• Banach, Stefan (1932).Théorie des Opérations Linéaires [Theory of Linear Operations](PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej.Zbl 0005.20901. Archived fromthe original(PDF) on 2014-01-11. Retrieved2020-07-11.

• Theory of continuous functions
• Operator theory
• Theorems in functional analysis

• CS1 French-language sources (fr)