Stone-Weierstrass theorem (complex version)

Theorem - Let X be a compact space and C(X) the algebraMathworldPlanetmath of continuous functionsMathworldPlanetmathPlanetmath X endowed with the sup norm . Let 𝒜 be a subalgebraPlanetmathPlanetmath of C(X) for which the following conditions hold:

  1. 1.

    x,yX,xy,f𝒜:f(x)f(y), i.e. 𝒜 separates points

  2. 2.

    1𝒜, i.e. 𝒜 contains all constant functions

  3. 3.

    If f𝒜 then f¯𝒜, i.e. 𝒜 is a self-adjoint ( subalgebra of C(X)

Then 𝒜 is dense in C(X).

Proof: The proof follows easily from the real version of this theorem (see the parent entry (

Let be the set of the real parts of elements f𝒜, i.e.


It is clear that contains (it is in fact equal) to the set of the imaginary parts of elements of 𝒜. This can be seen just by multiplying any function f𝒜 by -i.

We can see that 𝒜. In fact, Re(f)=f+f¯2 and by condition 3 this element belongs to 𝒜.

Moreover, is a subalgebra of 𝒜. In fact, since 𝒜 is an algebra, the productMathworldPlanetmathPlanetmath of two elements Re(f), Re(g) of gives an element of 𝒜. But since Re(f).Re(g) is a real valued function, it must belong to . The same can be said about sums and products by real scalars.

Let us now see that separates points. Since 𝒜 separates points, for every xy in X there is a function f𝒜 such that f(x)f(y). But this implies that Re(f(x))Re(f(y)) or Im(f(x))Im(f(y)), hence there is a function in that separates x and y.

Of course, contains the constant function 1.

Hence, we can apply the real version of the Stone-Weierstrass theorem to conclude that every real valued function in X can be uniformly approximated by elements of .

Let us now see that 𝒜 is dense in C(X). Let fC(X). By the previous observation, both Re(f) and Im(f) are the uniform limits of sequences {gn} and {hn} in . Hence,


Of course, the sequence {gn+ihn} is in 𝒜. Hence, 𝒜 is dense in C(X).

Title Stone-Weierstrass theorem (complex version)
Canonical name StoneWeierstrassTheoremcomplexVersion
Date of creation 2013-03-22 18:02:31
Last modified on 2013-03-22 18:02:31
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 6
Author asteroid (17536)
Entry type Theorem
Classification msc 46J10