 Solution

The odd-power family is also dense on $[1,2]$. It is $B=tA$, where $A$ is the even-power family in part (i). Multiplication by $t$ is an invertible bounded linear map on $C([1,2])$, with inverse division by $t$, since $t\ge 1$.

For an explicit approximation, take any $f\in C([1,2])$ and $ϵ>0$. The quotient $f(t)/t$ is continuous, so part (i) gives an even polynomial $a$ with $\Vert a-f/t{\Vert }_{\infty }<ϵ/2$. Then $ta$ belongs to the odd-power family andAlthough this family is not a subalgebra and does not contain the constant function exactly, it can approximate every continuous function, including constants. This proves even and odd polynomial density on a positive interval without incorrectly applying the unital algebra theorem directly to the odd family.