the distance between the endpoints of . If then . Schur's theorem is usually stated for curves, but John M. Sullivan has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different). Linear algebra In linear algebra, Schur’s theorem is referred to as either the triangularization of a square matrix with complex entries, or of a square matrix with real entries and real eigenvalues. Functional analysis In functional analysis and the study of Banach spaces, Schur's theorem, due to I. Schur, often refers to Schur's property, that for certain spaces, weak convergence implies convergence in the norm. Number theory In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values , then the set of primes that divide some member of S is infinite. See also Schur's lemma (from Riemannian geometry) References Herbert S. Wilf (1994). generatingfunctionology. Academic Press. Shiing-Shen Chern (1967). Curves and Surfaces in Euclidean Space. In Studies in Global Geometry and Analysis. Prentice-Hall. Issai Schur (1912). Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, Sitzungsberichte der Berliner Math. Further reading Dany Breslauer and Devdatt P. Dubhashi (1995). Combinatorics for Computer Scientists John M. Sullivan (2006). Curves of Finite Total Curvature. arXiv. Theorems in discrete mathematics Ramsey theory Additive combinatorics Theorems in combinatorics Theorems in differential geometry Theorems in linear algebra Theorems in functional analysis Computer-assisted proofs