Editing Stone–Weierstrass theorem (section)

== Stone–Weierstrass theorem, real version ==
The set {{math|C[''a'', ''b'']}} of continuous real-valued functions on {{math|[''a'', ''b'']}}, together with the supremum norm {{math|{{norm|''f''}} {{=}} sup<sub>''a'' ≤ ''x'' ≤ ''b''</sub> {{abs|''f'' (''x'')}}}} is a [[Banach algebra]], (that is, an [[associative algebra]] and a [[Banach space]] such that {{math|{{norm|''fg''}} ≤ {{norm|''f''}}·{{norm|''g''}}}} for all {{math| ''f'', ''g''}}). The set of all polynomial functions forms a subalgebra of {{math|C[''a'', ''b'']}} (that is, a [[linear subspace|vector subspace]] of {{math|C[''a'', ''b'']}} that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is [[Topology Glossary|dense]] in {{math|C[''a'', ''b'']}}.

Stone starts with an arbitrary compact Hausdorff space {{mvar|X}} and considers the algebra {{math|C(''X'', '''R''')}} of real-valued continuous functions on {{mvar|X}}, with the topology of [[uniform convergence]]. He wants to find subalgebras of {{math|C(''X'', '''R''')}} which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it ''[[separating set|separates points]]'': a set {{mvar|A}} of functions defined on {{mvar|X}} is said to separate points if, for every two different points {{mvar|x}} and {{mvar|y}} in {{mvar|X}} there exists a function {{mvar|p}} in {{mvar|A}} with {{math|''p''(''x'') ≠ ''p''(''y'')}}. Now we may state:

{{math theorem | name = Stone–Weierstrass theorem (real numbers) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space and {{mvar|A}} is a subalgebra of {{math|C(''X'', '''R''')}} which contains a non-zero constant function. Then {{mvar|A}} is dense in {{math|C(''X'', '''R''')}} [[if and only if]] it separates points.}}

This implies Weierstrass' original statement since the polynomials on {{math|[''a'', ''b'']}} form a subalgebra of {{math|C[''a'', ''b'']}} which contains the constants and separates points.

=== Locally compact version ===
A version of the Stone–Weierstrass theorem is also true when {{mvar|X}} is only [[locally compact]]. Let {{math|C<sub>0</sub>(''X'', '''R''')}} be the space of real-valued continuous functions on {{mvar|X}} that [[vanish at infinity]]; that is, a continuous function {{math| ''f'' }} is in {{math|C<sub>0</sub>(''X'', '''R''')}} if, for every {{math|''ε'' > 0}}, there exists a compact set {{math|''K'' ⊂ ''X''}} such that {{math| {{abs|''f''}}  < ''ε''}} on {{math|''X''&nbsp;\&nbsp;''K''}}. Again, {{math|C<sub>0</sub>(''X'', '''R''')}} is a [[Banach algebra]] with the [[supremum norm]]. A subalgebra {{mvar|A}} of {{math|C<sub>0</sub>(''X'', '''R''')}} is said to '''vanish nowhere''' if not all of the elements of {{mvar|A}} simultaneously vanish at a point; that is, for every {{mvar|x}} in {{mvar|X}}, there is some {{math| ''f'' }} in {{mvar|A}} such that {{math| ''f'' (''x'') ≠ 0}}. The theorem generalizes as follows:

{{math theorem | name = Stone–Weierstrass theorem (locally compact spaces) | math_statement = Suppose {{mvar|X}} is a ''locally compact'' Hausdorff space and {{mvar|A}} is a subalgebra of {{math|C<sub>0</sub>(''X'', '''R''')}}. Then {{mvar|A}} is dense in {{math|C<sub>0</sub>(''X'', '''R''')}} (given the topology of [[uniform convergence]]) if and only if it separates points and vanishes nowhere.}}

This version clearly implies the previous version in the case when {{mvar|X}} is compact, since in that case {{math|C<sub>0</sub>(''X'', '''R''') {{=}} C(''X'', '''R''')}}. There are also more general versions of the Stone–Weierstrass theorem that weaken the assumption of local compactness.<ref name=Willard>{{cite book |first=Stephen |last=Willard |title=General Topology |url=https://archive.org/details/generaltopology00will_0 |url-access=registration |page=[https://archive.org/details/generaltopology00will_0/page/293 293] |publisher=Addison-Wesley |year=1970 |isbn=0-486-43479-6 }}</ref>

=== Applications ===
The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result.
* If {{math| ''f'' }} is a continuous real-valued function defined on the set {{math|[''a'', ''b''] × [''c'', ''d'']}} and {{math|''ε'' > 0}}, then there exists a polynomial function {{mvar|p}} in two variables such that {{math|{{!}} ''f'' (''x'', ''y'') − ''p''(''x'', ''y'') {{!}} < ''ε''}} for all {{mvar|x}} in {{math|[''a'', ''b'']}} and {{mvar|y}} in {{math|[''c'', ''d'']}}.{{Citation needed|date=July 2018}}
* If {{mvar|X}} and {{mvar|Y}} are two compact Hausdorff spaces and {{math|''f'' : ''X'' × ''Y'' → '''R'''}} is a continuous function, then for every {{math|''ε'' > 0}} there exist {{math|''n'' > 0}} and continuous functions {{math| ''f''<sub>1</sub>, ...,  ''f<sub>n</sub>'' }} on {{mvar|X}} and continuous functions {{math|''g''<sub>1</sub>, ..., ''g<sub>n</sub>''}} on {{mvar|Y}} such that {{math|{{norm|''f'' − Σ&nbsp;''f<sub>i</sub> g<sub>i</sub>''}} < ''ε''}}. {{Citation needed|date=July 2018}}