Editing Stone–Weierstrass theorem (section)

== Bishop's theorem ==
Another generalization of the Stone–Weierstrass theorem is due to [[Errett Bishop]].  Bishop's theorem is as follows:<ref>{{citation|first=Errett|last=Bishop|author-link=Errett Bishop|title=A generalization of the Stone–Weierstrass theorem| journal=Pacific Journal of Mathematics|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pjm/1103037116&view=body&content-type=pdf_1| year=1961| volume=11| issue=3| pages=777–783| doi=10.2140/pjm.1961.11.777|doi-access=free}}</ref>

{{math theorem | name = Bishop's theorem | math_statement = Let {{mvar|A}} be a closed subalgebra of the complex [[Banach algebra]] {{math|C(''X'', '''C''')}} of continuous complex-valued functions on a compact Hausdorff space {{mvar|X}}, using the supremum norm. For  {{math|''S'' ⊂ ''X''}}  we write {{math|1=''A<sub>S</sub>'' = {''g{{!}}<sub>S</sub>'' : g &isin; ''A''}<nowiki/>}}. Suppose that {{math|''f''  ∈ C(''X'', '''C''')}} has the following property:
{{block indent | em = 1.5 | text = {{math| ''f'' {{!}}<sub>''S''</sub> ∈ ''A<sub>S</sub>''}} for every maximal set {{math|''S'' ⊂ ''X''}} such that all real functions of {{math|''A<sub>S</sub>''}} are constant.}}
Then {{math| ''f''  ∈ ''A''}}.}}

{{harvtxt|Glicksberg|1962}} gives a short proof of Bishop's theorem using the [[Krein–Milman theorem]] in an essential way, as well as the [[Hahn–Banach theorem]]: the process of {{harvtxt|Louis de Branges|1959}}.  See also {{harvtxt|Rudin|1973|loc=§5.7}}.