Let be a sequence of entire functions such that, for every , there exists . Suppose that is continuous and . Let be an entire function such that . Prove that over compact sets if and only if , for every .

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# It’s a matter of principles (of convergence)

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Let be a sequence of entire functions such that, for every , there exists . Suppose that is continuous and . Let be an entire function such that . Prove that over compact sets if and only if , for every .

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Recall that convergence over compact sets is given by the following metric on the set of entire functions:

where stands for the unit disk and are entire functions.

Suppose now that for every . Let be a compact set. Since by hypothesis the set is pointwise bounded, Montel’s theorem implies is precompact. Thus, every subsequence of admits a uniformly convergent subsequence. Since converges pointwise to the identity function, it converges uniformly over compact sets to the identity function. It follows easily that uniformly over compact sets.

For the other implication, fix and note that again by Montel every subsequence of admits a convergent subsequence to a certain function in . Hence, uniformly over . Since , there are open sets , such that and admits an inverse which is holomorphic. By continuity of the function , for a sufficiently small all the sets are contained in . Therefore, by precomposing with the holomorphic inverse of we see that coincides with the identity over . By the identity principle, it is for every . Since was arbitrary, the conclusion follows.

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