Exercise 19, § 4

Let $X$ be a compact space, and let $A$ be the normed $\mathbf{C}$-algebra $C(X; \mathbf{C})$. For each closed ideal $a$ of $A$, show that the relation $f \in a$ implies $\overline{f} \in a$ (note that $\overline{f}$ is the uniform limit of functions of the form $gf$). Extend the results of Exercise 13 to the algebra $A$.