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If $f_i$ are real analytic functions on $\mathbb R^n$ such that for arbitrary partial derivative index $\alpha \ge 0$, $f_i^\alpha \to {f^\alpha }$ uniformly, is it necessary that $f$ is an analytic function?

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    You say that $\alpha$ is just an index (and I guess, not a power) - then I wail to find how is is used in your statement.2011-12-06
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    Are you using $f^{\alpha}$ as notation for a partial derivative of $f$ of arbitrary order? If not, I, too, am puzzled as to how to understand your notation.2011-12-06
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    Sorry for the unclarity, I have clarified it.2011-12-06
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    In mathoverflow this question was discussed. http://mathoverflow.net/questions/53557/metric-on-the-space-of-real-analytic-functions2011-12-06

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