## Applied abstract analysis |

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### Contents

Introduction A Guide to the Reader | 1 |

Definitions and Examples | 5 |

The Field of Real Numbers | 7 |

Copyright | |

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associate ball B(x Banach space belongs bicontinuous bijective bounded functions Cauchy sequence Chapter closed sets closure cluster point co(K compact convex compact set compact space compact subset complete metric space conclude consequently Consider contained continuous linear continuous mapping convergent sequence converges pointwise converges uniformly convex function convex hull convex set convex subset countable deduce Definition denote distance equation equivalent example exists function defined Hence homeomorphism implies inequality inf f(x intersection interval isometry Let us show limit linear mapping locally compact lower semicontinuous lower semicontinuous function Moreover neighborhood nonempty closed subsets nonempty open normed space normed vector space obtain open ball open set pk(x Proof radius real numbers real-valued function Remark satisfies scalar product Section seminorm sequence of elements set of continuous solution space F subspace sup inf Suppose surjective topological upper semicontinuous xeE yeF