
“real: chapter_09” — 2011/5/22 — 23:21 — page 35 — #35
Sequences and Series of Functions 9-35
(i) If f ∈ A, then f defined by f (x) = f (x) ∈ A.
(ii)
A separates points on K.
(iii)
A vanishes at no point of K.
Then the uniform closure
B of A consists precisely of all complex
continuous functions on K.
Proof Let
A
R
be the algebra of all real continuous functions on K,
which belong to
A. For f ∈ A, we write f (x) = u(x) + iv(x) where
u, v ∈
A
R
and get u =
f +f
2
∈ A
R
.Ifx
1
= x
2
in K, then by hypothesis
(as in Step 2 of Theorem 9.4.19) there exists f ∈
A with f (x
1
) = 1 and
f (x
2
) = 0 or equivalently u(x
2
) = 0 = 1 = u(x
1
). This shows that A
R
separates points on K. Further ...