How can I show that the space of polynomials on $[0,1]$i.e,$P[0,1]$ is dense in $L^1[0,1]$ but not in $C[0,1]$?
Any help would be appreciated...
Problem on the density of $P[0,1]$
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functional-analysis
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0What denotes $P$ ? – 2017-01-28
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1probably polynomials, I have a question, is the norm in $C[0,1]$ the supremum norm? – 2017-01-28
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2Isn't the fact that $P[0,1]$ is dense in $C[0,1]$ the stone-weierstrass theorem? – 2017-01-28
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0@ Jorge Fernández Hidalgo:but that holds for sup norms .I have given $L^1$ norm – 2017-01-28
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2@vikash $\|\cdot\|_{L^1}≤\|\cdot\|_\infty$ so if $p_n\to f$ in sup norm so also in $L^1$ norm and $P$ is dense in $C$ with this norm also. – 2017-01-28
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2Aside from that since $C[0,1]$ is obviously a subspace of $L^1[0,1]$ if given the $L^1$ norm, $P$ being dense in $L^1$ implies $P$ being dense in $C[0,1]$. – 2017-01-28
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0@s.harp: thanks...now i get it – 2017-01-28