Editing Stone–Weierstrass theorem (section)

== Nachbin's theorem ==
Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold.<ref>{{citation|first=L.|last=Nachbin|title=Sur les algèbres denses de fonctions diffèrentiables sur une variété|journal=C. R. Acad. Sci. Paris|date=1949|volume=228|pages=1549–1551}}</ref> Nachbin's theorem is as follows:<ref>{{citation|first=José G.|last=Llavona| title=Approximation of continuously differentiable functions| date=1986| publisher=North-Holland| location=Amsterdam| isbn=9780080872414}}</ref>

{{math theorem | name = Nachbin's theorem | math_statement = Let {{mvar|A}} be a subalgebra of the algebra {{math|C<sup>∞</sup>(''M'')}} of smooth functions on a finite dimensional smooth manifold {{mvar|M}}. Suppose that {{mvar|A}} separates the points of {{mvar|M}} and also separates the tangent vectors of {{mvar|M}}: for each point ''m'' ∈ ''M'' and tangent vector ''v'' at the tangent space at ''m'', there is a ''f'' ∈ {{mvar|A}} such that d''f''(''x'')(''v'') ≠ 0. Then {{mvar|A}} is dense in {{math|C<sup>∞</sup>(''M'')}}.}}