Exercise 7.20

Following the hint, note that by the linearity of Riemann-integration, ${\int }_0^{1}f(x)P(x)\mathit{dx}=0$ for all polynomials $P$. By Theorem 7.26 there exists a sequence of polynomials $P_n$ which converge to $f$ uniformly on $[0,1]$. Since $[0,1]$ is compact, $f$ and each $P_n$ are bounded on $[0,1]$, so $f(x)P_n(x)$ converges uniformly to $f^2(x)$ by Exercise 7.2. By Theorem 7.16, $0={\int }_0^{1}f(x)P_n(x)\mathit{dx}$ must converge to ${\int }_0^1{f}^2(x)\mathit{dx}$, so this integral equals 0. Hence $f(x)=0$ on $[0,1]$ by Exercise 6.2.