Editing Stone–Weierstrass theorem (section)

=== 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}}