Mazur–Ulam theorem

The Mazur–Ulam theorem states that every surjective isometry between real normed spaces is affine: it preserves midpoints and differs from a linear isometry by a translation.

Isometry: A map between metric spaces that preserves the distance between every pair of points. The theorem starts with a surjective isometry and proves it must also preserve affine structure.
Normed vector space: A vector space equipped with a norm that assigns lengths to its vectors. The theorem applies to normed spaces, without requiring completeness.
Stanisław Mazur: A Polish mathematician whose work included functional analysis and the geometry of Banach spaces. Mazur co-authored the result that bears his name.
Isometry group: The group of distance-preserving bijections of a metric space under composition. The theorem identifies the form of every element when the space is a real normed space.
Midpoint: The point halfway between two points in an affine space. Preserving midpoints is the key intermediate conclusion of the theorem.
Metric space: A set equipped with a distance function satisfying the metric axioms. An isometry is defined through distances, even though the theorem’s spaces also have vector operations.
Stanisław Ulam: A Polish-American mathematician known for contributions to set theory, probability, and mathematical physics. Ulam co-authored the theorem with Mazur.
Banach–Stone theorem: A theorem characterizing certain linear isometries between spaces of continuous functions. It exemplifies how isometric structure can reveal linear structure in function spaces.
Affine map: A map that is a linear map followed by a translation. The theorem concludes that every surjective isometry has this form.
Norm (mathematics): A function assigning lengths to vectors and satisfying positivity, homogeneity, and the triangle inequality. Norm-induced distances supply the geometry that the theorem’s maps preserve.

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