1. Consider the following set of polynomials: $S := \{x^{2m}: m \geq 0\}$. Is there a non-zero element $f \in L^2([0,1])$ such that $\int\_0^1 fx^{2m} = 0$ for each $m \geq 0$? Note that the answer is negative if $[0,1]$ is replaced by $[-1,1]$.
2. More generally, given a set of polynomials, is there any standard procedure to check if its linear span is dense in $L^2(X)$?



How to determine if a given set of polynomials has dense linear span in $L^2([0,1])$?
You want a version of the classical Müntz–Szász theorem for the space $L^2([0,1])$ (which is, incidentally, the case considered initially by Szász). Here is a nice paper on the situation for $L^p([0,1])$ spaces, always for the span of monomials (with real exponents allowed)
<http://www.math.tamu.edu/~terdelyi/papers-online/Fullmuntz.pdf>


