= Even and odd polynomial density on a positive interval
{title2=$\overline{\mathbb C[t^2]}=\overline{t\mathbb C[t^2]}=C([a,b]),\quad0<a<b$}
On a positive compact interval, both even-power and odd-power complex <polynomial> families are uniformly dense in the continuous-function algebra. The even family is a conjugation-closed unital algebra separating points, since squaring is injective on the interval; the <Stone-Weierstrass theorem> proves its density. Equivalently use the <homeomorphism> $t\mapsto t^2$ and <polynomial> approximation on $[a^2,b^2]$. For the odd family, given a target $f$, first approximate $f/t$ by an even <polynomial> $p(t^2)$. Then $tp(t^2)$ is odd and its error is at most $b$ times the first error. Division is continuous because the interval avoids zero. Thus the odd family can be dense even though it is not a unital <subalgebra>.
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