##  [Stone–Weierstrass Theorem](/index.php/stone-weierstrass-theorem) 

  ##  [Stone–Weierstrass Theorem](https://mathlogic.quantumdictionary.io/stone-weierstrass-theorem-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/index.php/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A characterization asserting that a subalgebra A of C(X), the real-valued continuous functions on a compact Hausdorff space X, is dense in the uniform topology if and only if A contains the constants and separates the points of X (with an added *-closure condition in the complex-valued case).

 

 

 

 

 





 

 



 ##  [Stone–Weierstrass Theorem](https://puremath.quantumdictionary.io/stone-weierstrass-theorem-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/index.php/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A theorem giving conditions under which a subalgebra A of C(X) (continuous real- or complex-valued functions on a compact Hausdorff space X) is uniformly dense in C(X): typically A must contain the constants and separate points, and in the complex case be closed under complex conjugation (a *‑subalgebra).