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In page Stone–Weierstrass theorem:

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The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result.

  • If  f  is a continuous real-valued function defined on the set [a, b] × [c, d] and ε > 0, then there exists a polynomial function p in two variables such that | f (x, y) − p(x, y) | < ε for all x in [a, b] and y in [c, d].[citation needed]
  • If X and Y are two compact Hausdorff spaces and f : X × Y → R is a continuous function, then for every ε > 0 there exist n > 0 and continuous functions  f1, ...,  fn  on X and continuous functions g1, ..., gn on Y such that ‖f − Σ fi gi‖ < ε.[citation needed]