Can a sequence of polynomials in a translation invariant linear space (of polynomials, of course) point-wise converge to a polynomial which is not included in that space? In one variable this is impossible.
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$\begingroup$ A sequence of polynomials converges point-wise to a polynomial if and only if it converges coefficient-wise to it. A linear space of polynomials is translation invariant if and only if it is invariant under partial differentiation. $\endgroup$– user16456Commented Jul 16, 2011 at 6:37
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1$\begingroup$ The first claim is true only if the degrees are uniformly bounded. $\endgroup$– fedjaCommented Jul 16, 2011 at 12:43
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1$\begingroup$ (precisely, point-wise convergence implies coefficient-wise convergence, but not the opposite, as the example $x^n$ shows). $\endgroup$– Pietro MajerCommented Jul 18, 2011 at 12:04
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$\begingroup$ (Pietro Majer) Yeah, sorry. $\endgroup$– user16456Commented Jul 19, 2011 at 3:27
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1$\begingroup$ Coefficients are linear combinations of derivatives, derivatives are linear combinations of differences, differences are linear combinations of function values - hence point-wise convergence does imply coefficient-wise convergence. $\endgroup$– user16456Commented Jul 21, 2011 at 3:04
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