In $L_2$ space, only use domain [0, 1]. Prove, for any $C^2$ function F in $L_2$, there exits a sequence of polynomials ($f_k$) such that $f^{(r)}_k$ tends to $F^{(r)}$ for r=0,1,2 uniformly. Note: $f^{(r)}_k$ means the $r^{th}$ derivative of $f_k$.
functional analysis question: extended version of completion of $L_2$ space
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functional-analysis
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1To everybody: please do refrain from trying to edit somebody else's question to ask one of your own! – 2012-10-16
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0What do you mean by a function $F$ $C^2$ in $L_2$? If $f$ is $C^2$, by Stone-Weierstrass, a sequence of polynomial converges uniformly on $[0,1]$ to $f''$. We can use this sequence to get an approximation of $f'$ and $f$. – 2012-10-16
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0thanks for your answer. It is right. I confused myself before. – 2012-10-17