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Show that if $f$ is a continuous function on the interval $[a , b]$ and if $\int_a^b f(x) p(x)dx = 0$ for every polynomial $p$ then $f$ must be the zero function.

Attempt:

By the Weierstrass Approximation Theorem, we know there is a sequence $\{p_j\}$ such that $p_j \rightarrow f$ uniformly on $[a,b]$. (The Theorem states: "Let $f$ be a continuous function on an interval $[a,b]$. Then there is a sequence of polynomias $p_j(x)$ with the property that the sequence $p_j$ converges uniformly on $[a,b]$ to $f$.")

Then since $p_j$ is integrable on $[a,b]$ and since $p_j \rightarrow f$ uniformly on $[a,b]$. Then we know $f$ is integrable on $[a,b]$ and $\lim_{j \to \infty} \int_a^b p_{j}(x) dx = \int_a^b f(x) dx $.

But how do I know $f(x) = 0$ ?

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    @David: Thanks for that example. I deleted my answer.2012-03-03
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    Hint: $\int_a^b f(x) p_j(x) \, dx = 0$ by assumption, and by uniform convergence $\int_a^b f(x)^2 \, dx = 0$. Can you finish from here?2012-03-03

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