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Let $E=\mathbb{R}[X]$

We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$.

Help me to study the continuity of the linear form $f_m\colon\, P \to P_{[m]}$ ($P_{[m]}$ being the coefficient of $x^m$ in $P$) for some positive integer $m$.

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This isn't a very difficult problem. What have you tried? Where are you getting stuck? It is enough to prove continuity at zero (i.e. if $P_k$ is a sequence converging to zero, in the $N(\cdot)$ norm, then for each $m$, $f_m(P_k)$ converges to zero). To prove this, prove that if $M(P_k)$ denotes the leading coefficient, and if $N(P_k) \leq \epsilon$, then $|M(P_k)| \leq \epsilon/\mathrm{deg}(P_k)!$. So $M(P_k)$ converges to zero. Use this to prove that the coefficient of the $x^{\mathrm{deg(P_k) - 1}}$ term must also converge to zero, and so on... –  William Oct 21 '12 at 3:50
    
actually for m=1 there is discontinuity –  mathfan Oct 22 '12 at 20:59

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