Editing Stone–Weierstrass theorem (section)

=== 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>