Is the set of all polynomial closed in the $ C[a,b] $ space?

Question

Is the set of all polynomial closed in the $ C[a,b] $ space ?

Answer 1

No. If we have the norm $||f||:= \max\{|f(x)|: x \in [a,b]\}$ on $C[a,b]$ and if we denote by $P$ the set of all poynomials, then $P$ is a subspace of $C[a,b]$ with

$$\overline{P}=C[a,b].$$

This is the Approximation Theorem of Weierstraß

$\overline{P}$ denotes the closure of $P$ in $(C[a,b],||*||) $ .

Answer 2

On the contrary, for the $\mathcal C^\infty$ norm (uniform convergence norm), Stone-Weierstrass' theorem asserts the set of polynomial functions is dense in $\mathcal C \bigl([a,b]\bigr)$.