Define $A$ to be the family of polynomials containing only even powers of $x$. Is $A$ dense in $C[0,1]$?

Question

Define $A$ to be the family of polynomials containing only even powers of $x$. Is $A$ dense in $C[0,1]$?

How to approach these kind of problems? Would the same technique apply to $B$, the family of polynomials containing only odd powers of $x$ in $C[-1,1]$?

Answer 1

Here is a simple argument which does not require Stone-Weierstrass Theorem: if $f$ is continuous so is $f(\sqrt x)$. If $|f(\sqrt x)-p(x)|<\epsilon$ then $|f(x)-p(x^{2})|<\epsilon$ and $p(x^{2})$ has only even powers of $x$. For the second part note that if a sequence of polynomials with only odd powers of $x$ converges pointwise on $[-1,1]$ then the limit is an odd function. So functions (like the constant function 1) which are not odd cannot be approximated by polynomials with only odd powers.

Answer 2

Stone-Weierstrass Theorem says that if $A$ is an algebra of $C(X)$ that separates points in $X$ and contains the constant functions, then $A$ is dense in $C(X)$, here $X$ is compact.

Realizing to the set $A$ and $X=[0,1]$ in question, even powers of polynomials contain those of constants, and clearly it is an algebra which separates points, consider for example, $x^{2}$, this is an one-to-one function.