of the Weierstrass approximation theorem is that this subalgebra is dense in . Stone starts with an arbitrary compact Hausdorff space and considers the algebra of real-valued continuous functions on , with the topology of uniform convergence. He wants to find subalgebras of which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set of functions defined on is said to separate points if, for every two different points and in there exists a function in with . Now we may state: This implies Weierstrass' original statement since the polynomials on form a subalgebra of which contains the constants and separates points. Locally compact version A version of the Stone–Weierstrass theorem is also true when is only locally compact. Let be the space of real-valued continuous functions on that vanish at infinity; that is, a continuous function is in if, for every , there exists a compact set such that on . Again, is a Banach algebra with the supremum norm. A subalgebra of is said to vanish nowhere if not all of the elements of simultaneously vanish at a point; that is, for every in , there is some in such that . The theorem generalizes as follows: This version clearly implies the previous version in the case when is compact, since in that case . There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness. Applications The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result. If is a continuous real-valued function defined on the set and , then there exists a polynomial function in two variables such that for all in and in . If and are two compact Hausdorff spaces and is a continuous function, then for every there exist and continuous functions on and continuous functions on such that . The theorem has many other applications to analysis, including: Fourier series: The set of linear combinations of functions is dense in , where we identify the endpoints of the interval to obtain a circle. An important consequence of this is that the are an orthonormal basis of the space of square-integrable functions on . Stone–Weierstrass theorem, complex version Slightly more general is the following theorem, where we consider the algebra of complex-valued continuous functions on the compact space , again with