# Tag Archives: Stone-Weierstrass

## Analysis example: int f(x) x^n dx = 0 implies f = 0

Problem. Let $$f$$ be continuous on $$[0,~1]$$ and let $\int_{0}^{1} f(x) x^n ~ dx =0$ for every nonnegative integer $$n .$$ Prove that $$f=0$$ on $$[0,~1] .$$ We first observe two lemmas. These lemmas are for beginners, so you may skip them. Lemma 1. Under the same condition, $\int_{0}^{1} f(x) p(x) ~ dx =0$ holds… Read More »