While studying for an exam, I came across the following problem.
Suppose $[a,b]\subset\mathbb{R}$ and $f:[a,b]\rightarrow\mathbb{R}$ is a strictly increasing non-vanishing continuous function. Suppose $g:[a,b]\rightarrow\mathbb{R}$ is a continuous function which is orthogonal to all powers of $f$, i.e. is such that $\int_a^bg(x)(f(x))^n\,dx=0$ (Riemann integrals) for $n=1,2,3,\ldots$. Show that $g(x)=0$ for all $x\in[a,b]$.
Since I could not think of how to solve this problem, I considered something simpler.
I know that if $f(x)=x$, then we have $\int_a^bg(x)x^n\,dx=0$ for all $n$. Applying the Weierstrass Approximation Theorem to $g$, we obtain a sequence of polynomials $(p_n)$ such that $p_n\rightarrow g$ uniformly. Therefore, by linearity of the integral, we have $\int_a^bg(x)p_n(x)\,dx=0$ for all $n$. Thus $\lim_{n\rightarrow\infty}\int_a^bg(x)p_n(x)\,dx=\int_a^b(g(x))^2\,dx=0.$ Since $g$ is continuous, the result follows.
But, obviously, this is not an answer to the stated question...
Any help would be greatly appreciated. Thanks in advance.