= Uniform closure of a function algebra
= Uniform closure
{synonym}
For an <algebra> $A\subseteq C(K,\mathbb F)$ on a <compact Hausdorff space> $K$, with $\mathbb F=\mathbb R$ or $\mathbb C$, its <uniform closure> consists of the <uniform limits> of sequences from $A$. It is a closed <algebra> in the <supremum norm>, since uniformly convergent sequences are uniformly bounded and
$$
\|f_ng_n-fg\|_\infty\leq\|f_n\|_\infty\|g_n-g\|_\infty+\|f_n-f\|_\infty\|g\|_\infty.
$$
For a \b[real] <algebra> containing the constants and separating points, the <Stone-Weierstrass theorem> makes this closure all of $C(K,\mathbb R)$. For a \b[complex] <algebra>, closure under <complex conjugation> is additionally sufficient; separation and constants alone are not enough.<Basic Analysis: Introduction to Real Analysis, Volume II by Jiří Lebl>[section11.7, Theorem11.7.16, complex Stone–Weierstrass]{ref}
For example, complex <polynomials> in $z$ on the closed <unit disk> contain constants and separate points, but their <uniform closure> is the <disk algebra>, not all continuous <functions> on that disk. They cannot approximate $\overline z$ uniformly: uniform limits of these <holomorphic functions> are <holomorphic> in the interior, while $\overline z$ is not.
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