## A Guide to Advanced Real AnalysisThis concise guide to real analysis covers the core material of a graduate level real analysis course. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form. The prerequisite is a familiarity with classical real-variable theory. |

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### Contents

Topology | 5 |

General Theory | 21 |

Constructions and Special Examples | 41 |

Rudiments of Functional Analysis | 63 |

Function Spaces | 75 |

Bibliography | 101 |

### Common terms and phrases

a-algebra a-finite absolutely continuous algebra arbitrary ball Banach space bijection Borel measure bounded linear called Cauchy CC(X closed sets compact Hausdorff space compact sets complement complex measure continuous functions Corollary countable decomposition defined denote dense differentiation distributions E C X easy Euclidean space everywhere example fact finite follows Fourier series Fourier transform Hausdorff space Henstock-Kurzweil integrable Hilbert space inequality integrable function interval LCH space Lebesgue measure linear functional linear map Ll(p locally integrable Lp norm Lp spaces Mathematical measurable functions measurable sets measure space metric space Moreover neighborhood nonempty set normed vector space notion open sets orthonormal outer measure outer regularity pointwise convergence proof Proposition regular Borel measure result Ross Honsberger seminorms sequence simple functions subsets subspace Suppose Theorem 6.2 theory tion topological space unions unique