Editing Stone–Weierstrass theorem (section)

== Stone–Weierstrass theorem, complex version ==
Slightly more general is the following theorem, where we consider the algebra  <math>C(X, \Complex)</math> of complex-valued continuous functions on the compact space <math>X</math>, again with the topology of uniform convergence. This is a [[C*-algebra]] with the *-operation given by pointwise [[complex conjugation]].

{{math theorem | name = Stone–Weierstrass theorem (complex numbers) | math_statement = Let <math>X</math> be a compact Hausdorff space and let <math>S</math> be a [[separating set|separating subset]] of <math>C(X, \Complex)</math>. Then the complex [[unital algebra|unital]] [[*-algebra]] generated by <math>S</math> is dense in <math>C(X, \Complex)</math>.}}

The complex unital *-algebra generated by <math>S</math> consists of all those functions that can be obtained from the elements of <math>S</math> by throwing in the constant function {{math|1}} and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times.

This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, <math>f_n\to f</math>, then the real parts of those functions uniformly approximate the real part of that function, <math>\operatorname{Re}f_n\to\operatorname{Re}f</math>, and because for real subsets, <math>S\subset C(X,\Reals)\subset C(X,\Complex),</math> taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated.

As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.

The following is an application of this complex version.
* [[Fourier series]]: The set of linear combinations of functions {{math|''e<sub>n</sub>''(''x'') {{=}} ''e''<sup>2''πinx''</sup>, ''n'' ∈ '''Z'''}} is dense in {{math|C([0, 1]/{0, 1})}}, where we identify the endpoints of the interval {{math|[0, 1]}} to obtain a circle.  An important consequence of this is that the {{math|''e<sub>n</sub>''}} are an [[orthonormal basis]] of the space [[Lp space|{{math|L<sup>2</sup>([0, 1])}}]] of [[square-integrable function]]s on {{math|[0, 1]}}. {{Citation needed|date=May 2024}}