## Real AnalysisThis is a course in real analysis directed at advanced undergraduates and beginning graduate students in mathematics and related fields. Presupposing only a modest background in real analysis or advanced calculus, the book offers something to specialists and non-specialists alike, including historical commentary, carefully chosen references, and plenty of exercises. |

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By far the best real analysis text I've ever used. Especially good for those interested in pursuing functional analysis.

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Carothers has a wonderful style of writing that makes difficult concepts seem very understandable to the beginning graduate student. It is comprehensive, with a slant towards functional analysis. Exercises are excellent. Much easier read than classics like Royden.

### Contents

III | 3 |

IV | 14 |

V | 17 |

VI | 18 |

VII | 25 |

VIII | 31 |

IX | 34 |

X | 36 |

LII | 202 |

LIII | 210 |

LIV | 212 |

LV | 214 |

LVI | 215 |

LVII | 221 |

LVIII | 225 |

LIX | 232 |

XI | 37 |

XII | 39 |

XIII | 43 |

XIV | 45 |

XV | 49 |

XVI | 51 |

XVII | 53 |

XVIII | 60 |

XIX | 62 |

XX | 63 |

XXI | 69 |

XXII | 73 |

XXIII | 78 |

XXIV | 87 |

XXV | 89 |

XXVI | 92 |

XXVII | 97 |

XXVIII | 102 |

XXIX | 106 |

XXX | 108 |

XXXI | 114 |

XXXII | 120 |

XXXIII | 126 |

XXXIV | 128 |

XXXV | 131 |

XXXVI | 136 |

XXXVII | 137 |

XXXVIII | 139 |

XXXIX | 143 |

XL | 150 |

XLI | 153 |

XLII | 160 |

XLIII | 162 |

XLIV | 170 |

XLV | 176 |

XLVI | 178 |

XLVII | 183 |

XLVIII | 185 |

XLIX | 188 |

L | 194 |

LI | 201 |

LX | 234 |

LXI | 239 |

LXII | 242 |

LXIII | 244 |

LXIV | 250 |

LXV | 254 |

LXVI | 257 |

LXVII | 258 |

LXVIII | 261 |

LXIX | 263 |

LXX | 268 |

LXXI | 274 |

LXXII | 277 |

LXXIII | 283 |

LXXIV | 289 |

LXXV | 292 |

LXXVI | 293 |

LXXVII | 296 |

LXXVIII | 302 |

LXXIX | 304 |

LXXX | 306 |

LXXXI | 310 |

LXXXII | 312 |

LXXXIII | 314 |

LXXXIV | 322 |

LXXXV | 328 |

LXXXVI | 333 |

LXXXVII | 335 |

LXXXVIII | 337 |

LXXXIX | 342 |

XC | 350 |

XCI | 352 |

XCII | 356 |

XCIII | 359 |

XCIV | 370 |

XCV | 377 |

379 | |

XCVII | 395 |

XCVIII | 397 |

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### Common terms and phrases

algebra Borel set bounded sequence bounded variation BV[a Cantor function Cantor set Cauchy sequence Chapter closed sets complete consider contains continuous function Convergence Theorem converges in measure converges pointwise converges uniformly Corollary define definition denote dense derived numbers differentiable easy equicontinuous equivalent example Exercise exists fact finite fn(x follows Fourier series Given hence Hint homeomorphic infinite integrable functions isometry lattice Lebesgue integral Lebesgue measurable Lebesgue's Lemma Let f limit limn Lipschitz measurable functions measurable sets measure zero monotone nonempty nonnegative normed vector space notation null set open intervals open sets pairwise disjoint particular partition Prove real numbers real-valued functions recall Riemann integrable Riesz satisfies sequence xn simple functions step function subalgebra subset subspace supn suppose totally bounded trig polynomial uncountable uniform convergence uniformly continuous