## Continuous Functions of Vector Variables |

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### Contents

Sequences in Normed Spaces | 33 |

Limits and Continuity in Normed Spaces | 55 |

Characteristics of Continuous Functions | 85 |

Copyright | |

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accumulation point algebra arc-connected Assume that f ball Bolzano-Weierstrass theorem boundary bounded magnification bounded set Cantor set Cauchy sequence Cauchy's Cauchy's criterion closed and bounded closed sets closure point compact set Compare Exercise component conclude contained continuous function convex coordinates defined disconnected disjoint distance domain dot product equation equivalent Example exists extreme value property f is continuous Figure finite dimension finite limit finite-dimensional graph Hence implies infinite inner product space interior intermediate value intersection interval isolated point line joining maxnorm metric multiple neighborhood N(b nonempty nonnegative nonzero normed linear space normed space normed vector space open set orthogonal polynomial proj(x Proof Prove Theorem Pythagorean real number says Section 4.3 segment sequentially compact Show Similarly sublimit subspace Suppose Theorem 2.5 triangle inequality unbounded uniformly continuous union vector function vector space zero