Uniform closure of polynomials

Question

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.

Answer 1

The full quote is:

If $\partial\mathbb{D}=\{z\in\mathbb{C}:|z|=1\}$, let $B=$ the uniform closure of the polynomials in $C(\partial\mathbb{D})$.

This means: consider the set of all continuous functions on $\partial\mathbb{D}$, equipped with the uniform norm $\|f\|=\sup_{\partial \mathbb{D}}|f|$. This space is denoted by $C(\partial\mathbb{D})$. Polynomials form a subset of this space. The closure of this subspace is the set of all limits of uniformly convergent sequences of polynomials.

For example, the function $z\mapsto \exp(z)$ belongs to this closure, because the sequence of partial sums of the exponential function converges uniformly on the unit circle. On the other hand, the conjugation $z\mapsto \bar z$ does not belong to the closure, which can be proved with the argument principle (its values move the wrong way around $0$ as we trace the circle).