## Introductory Real Analysis |

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### Contents

SET THEORY Page | 1 |

METRIC SPACES Page | 37 |

TOPOLOGICAL SPACES Page | 78 |

Copyright | |

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arbitrary Banach space base belongs bounded called choose Clearly closed compact complete concept condition consider consisting construction contains continuous linear converges convex COROLLARY corresponding countable defined DEFINITION denoted dense derivative difference differentiable elements equal equation equipped equivalent Euclidean space everywhere Example exists extension fact Figure finite fixed follows function f give given hence holds implies inequality infinite integral intersection interval introduced least Lebesgue lemma limit linear functional linear operator mapping mean measure metric space Moreover neighborhood normed linear space Note obviously open set ordered particular positive Problem Proof properties Prove recall Remark respect Riemann integral satisfying sense sequence sphere subset subspace sufficiently Suppose taking Theorem topological space union unique unit values variation vectors weak zero