Banach–Stone theorem

The Banach–Stone theorem states that a surjective linear isometry between spaces of continuous scalar-valued functions on compact Hausdorff spaces forces the spaces to be homeomorphic.

Continuous function space: A vector space of continuous functions on a topological space, often equipped with a norm. The theorem starts with isometries between these spaces, usually carrying the supremum norm.
Banach space: A complete normed vector space. Continuous-function spaces with the supremum norm are Banach spaces when their domains are compact.
Mazur–Ulam theorem: A theorem stating that every surjective isometry between real normed spaces is affine. It explains why surjective isometries have affine structure, while Banach–Stone additionally recovers domain topology.
Gelfand representation: A representation of a commutative unital C*-algebra as continuous functions on a compact Hausdorff space. It generalizes the theorem’s principle that algebraic or geometric structure can reveal an underlying space.
Supremum norm: The norm of a bounded function given by the supremum of its absolute value. This norm makes the geometry of each function space encode properties of its domain.
Compact Hausdorff space: A topological space that is compact and in which distinct points have disjoint neighborhoods. These are the domains whose topology the theorem recovers from their continuous functions.
Banach–Stone theorem for locally compact spaces: A Banach–Stone type result relating spaces of continuous functions vanishing at infinity to locally compact Hausdorff spaces. It extends the compact-space setting by replacing C(X) with functions vanishing at infinity.
Commutative C*-algebra: A C*-algebra in which multiplication is commutative. Continuous functions on compact spaces form the central examples linking topology with operator algebra.
Extreme point: A point of a convex set that is not a nontrivial convex combination of other points in the set. Extreme points of dual unit balls help identify evaluation functionals at points of the compact spaces.
C(X): The space of continuous scalar-valued functions on a topological space X. The theorem is commonly stated for C(X) and C(Y) with their supremum norms.

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