Editing Stone–Weierstrass theorem (section)

== Stone–Weierstrass theorem, quaternion version ==
Following {{harvtxt|Holladay|1957}}, consider the algebra {{math|C(''X'', '''H''')}} of quaternion-valued continuous functions on the compact space {{mvar|X}}, again with the topology of uniform convergence. 

If a quaternion {{math|''q''}} is written in the form <math display=inline>q = a + ib + jc + kd</math>
*its scalar part {{math|''a''}} is the real number <math>\frac{q - iqi - jqj - kqk}{4}</math>.
Likewise 
*the scalar part of {{math|−''qi''}} is {{math|''b''}} which is the real number <math>\frac{-qi - iq + jqk - kqj}{4}</math>.
*the scalar part of {{math|−''qj''}} is {{math|''c''}} which is the real number <math>\frac{-qj - iqk - jq + kqi}{4}</math>.
*the scalar part of {{math|−''qk''}} is {{math|''d''}} which is the real number <math>\frac{-qk + iqj - jqk - kq}{4}</math>.

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