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In the space of polynomials with degree less than $N+1$: $\operatorname{span}\{ 1,x,\ldots,x^N\}$ defined on $[a,b]$, if a sequence has a uniform convergent limit(under maximum norm in continuous function space).

1) Is the limit a polynomial?

2) Is the limit polynomial's degree less than $N+1$?

3) Do the coeffcients of polynomial sequence converge?

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    I only know that if each coefficient converges, the polynomial sequence will converge uniformly.2012-08-07

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The answer is pretty simple with the right tools apparently: Let $\| P\|:= \sup_{0\leq k\leq N} |a_k|$ where $P(x)= \sum_{k=1}^{N} a_kx^k$. This is a norm in a finite dimensional space (a basis, as you give is $\{ 1,x,..., x^N\}$) but since all norms are equivalent in finite dimensional spaces, this norm (convergence in which implies convergence of the coefficients) is equivalent to the restriction of the maximum norm. The fact that the limit is a polynomial is a consequence of the fact that finite dimensional normed spaces are complete.