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Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3
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Small ball Gaussian probabilities with moving center
This is not a full answer but maybe of interest. Given a sequence of $x_n$ with $\|x_n\|_E\to \infty$ we can always find a sequence of $\delta_n\to 0$ so that
$$\lim_{n\to\infty}\frac{\mu(B_{\delta_n} …