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?
The limit of analytic functions
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calculus
analysis
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0You 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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0Are 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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0Sorry for the unclarity, I have clarified it. – 2011-12-06
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1In mathoverflow this question was discussed. http://mathoverflow.net/questions/53557/metric-on-the-space-of-real-analytic-functions – 2011-12-06