## Approximation Theory in the Central Limit Theorem: Exact Results in Banach Spaces~Et mai . ... , si j'avait su comment en revenir. One service mathematics has rendered the human race. It has put common sense back je n'y serais point aIIe.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent: therefore we may be sense' . able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series. |

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### Contents

Some Questions of NonLinear Analysis | 11 |

The Central Limit Theorem in Banach Spaces | 44 |

Gaussian Measure of eStrip of Some Sets | 62 |

Copyright | |

2 other sections not shown

### Common terms and phrases

apply approximation assume balls Banach space Bentkus bounded called central limit theorem centre chapter choose condition consider construct continuous COROLLARY covariance operator defined DEFINITION denote density depends derivative differentiable distance distribution easily easy equality equivalent estimate takes place example exists a constant fact finite finite-dimensional following estimate takes following inequality formula Fréchet function f Gaussian measure given Hence Hilbert space holds implies inclusion independent inequality infinitely integral interval LEMMA mapping mathematical mean method metric necessary norm noted obtain particular present probability problem Proof properties PROPOSITION proved quantity random elements rate of convergence relation respect result satisfied satisfies the conditions sense separable sequence smooth space H statements sufficient summands sums Supplement symmetric taking into account uniform uniformly V.Yu write zero