Editing Stone–Weierstrass theorem (section)

== Lattice versions ==
Let {{mvar|X}} be a compact Hausdorff space. Stone's original proof of the theorem used the idea of [[lattice (order)|lattices]] in {{math|C(''X'', '''R''')}}.  A subset {{mvar|L}} of {{math|C(''X'', '''R''')}} is called a [[lattice (order)|lattice]] if for any two elements {{math| ''f'', ''g'' ∈ ''L''}}, the functions {{math|max{ ''f'', ''g''}, min{ ''f'', ''g''} }}also belong to {{mvar|L}}. The lattice version of the Stone–Weierstrass theorem states:

{{math theorem | name = Stone–Weierstrass theorem (lattices) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space with at least two points and {{mvar|L}} is a lattice in {{math|C(''X'', '''R''')}} with the property that for any two distinct elements {{mvar|x}} and {{mvar|y}} of {{mvar|X}} and any two real numbers {{mvar|a}} and {{mvar|b}} there exists an element {{math| ''f''  ∈ ''L''}} with {{math| ''f'' (''x'') {{=}} ''a''}} and {{math| ''f'' (''y'') {{=}} ''b''}}. Then {{mvar|L}} is dense in {{math|C(''X'', '''R''')}}.}}

The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the [[absolute value]] {{math|{{!}} ''f'' {{!}}}} which in turn can be approximated by polynomials in {{math| ''f'' }}. A variant of the theorem applies to linear subspaces of {{math|C(''X'', '''R''')}} closed under max:<ref>{{citation|first1=E|last1=Hewitt|author-link=Edwin Hewitt | first2=K|last2=Stromberg | title=Real and abstract analysis| year=1965|publisher=Springer-Verlag | at = Theorem 7.29}}</ref>

{{math theorem | name = Stone–Weierstrass theorem (max-closed) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space and {{mvar|B}} is a family of functions in {{math|C(''X'', '''R''')}} such that
# {{mvar|B}} separates points.
# {{mvar|B}} contains the constant function 1.
# If {{math| ''f''  ∈ ''B''}} then {{math|''af''  ∈ ''B''}} for all constants {{math|''a'' ∈ '''R'''}}.
# If {{math| ''f'',  ''g'' ∈ ''B''}}, then {{math| ''f''  + ''g'', max{ ''f'', ''g''} ∈ ''B''}}.
Then {{mvar|B}} is dense in {{math|C(''X'', '''R''')}}.}}

More precise information is available:

:Suppose {{mvar|X}} is a compact Hausdorff space with at least two points and {{mvar|L}} is a lattice in {{math|C(''X'', '''R''')}}. The function {{math|''φ'' ∈ C(''X'', '''R''')}} belongs to the [[closure (topology)|closure]] of {{mvar|L}} if and only if for each pair of distinct points ''x'' and ''y'' in {{mvar|X}} and for each {{math|''ε'' > 0}} there exists some {{math| ''f''  ∈ ''L''}} for which {{math|{{!}} ''f'' (''x'') − ''φ''(''x''){{!}} < ''ε''}} and {{math|{{!}} ''f'' (''y'') − ''φ''(''y''){{!}} < ''ε''}}.