two points and is a lattice in . The function belongs to the closure of if and only if for each pair of distinct points x and y in and for each there exists some for which and . Bishop's theorem Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows : gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem : the process of . See also . Nachbin's theorem Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold . Nachbin's theorem is as follows : Editorial history In 1885 it was also published in an English version of the paper whose title was On the possibility of giving an analytic representation to an arbitrary function of real variable. According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series". See also Müntz–Szász theorem Bernstein polynomial Runge's phenomenon shows that finding a polynomial such that for some finely spaced is a bad way to attempt to find a polynomial approximating uniformly. A better approach, explained e.g. in , p. 160, eq. (51) ff., is to construct polynomials uniformly approximating by taking the convolution of with a family of suitably chosen polynomial kernels. Mergelyan's theorem, concerning polynomial approximations of complex functions. Notes References . . Jan Brinkhuis & Vladimir Tikhomirov (2005) Optimization: Insights and Applications, Princeton University Press . . . . . . . Historical works The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II). Erste Mitteilung (part 1) pp. 633–639, Zweite Mitteilung (part 2) pp. 789–805. Important historical works of Stone include: . ; 21 (5), 237–254. External links Theory of continuous functions Theorems in analysis Theorems in approximation theory 1885 in mathematics 1937 in mathematics