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Let $X$ denotes the span {$x^n:n \ge1 $}. Is it true that $X $ is dense in $L^1([0,1])?.$ I showed that $X$ is dense in the space of continuous functions that vanishes at zero. I also know space of continuous functions with compact support is dense in $L^p$ space. I guess it not true but my friend told me it is true. Anybody's help would be appreciated

mle
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Toeplitz
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Hint: given any continuous function on $[0,1]$, you can approximate it in the $L^1$ norm by continuous functions that vanish at $0$.

Robert Israel
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  • Thanks for the hint but I am still confused could you please elaborate little bit.. – Toeplitz Mar 23 '14 at 23:03
  • Given any $f \in L^1$ and $\epsilon > 0$, there is a continuous function $g$ on $[0,1]$ with $|f - g|1 < \epsilon/3$. There is a continuous function $h$ on $[0,1]$ with $h(0) = 0$ and $|g - h|_1 < \epsilon/3$. There is $p \in X$ with $|h - p|_1 \le |h - p|\infty < \epsilon/3$... – Robert Israel Mar 23 '14 at 23:42
  • By the help of your hint, i also did exactly same, thank you it helped me to verify my work... – Toeplitz Mar 24 '14 at 00:01