## Linear Operators: General theory |

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Page 19

The metric topology in X is the smallest topology containing the spheres . The set

X , with its metric topology , is called a

such that there exists a metric function whose topology is the same as the original

...

The metric topology in X is the smallest topology containing the spheres . The set

X , with its metric topology , is called a

**metric space**. If X is a topological spacesuch that there exists a metric function whose topology is the same as the original

...

Page 20

A sequence { an } in a

0 . If every Cauchy sequence is convergent , a

complete . The next three lemmas are immediate consequences of the definitions

.

A sequence { an } in a

**metric space**is a Cauchy sequence if limmin elam , an ) =0 . If every Cauchy sequence is convergent , a

**metric space**is said to becomplete . The next three lemmas are immediate consequences of the definitions

.

Page 459

13 Let X be a B - space , and let K * be a bounded X - closed convex subset of X *

. Show that K * has a continuous tangent functional at each point of a dense

subset of its boundary . 14 DEFINITION . A point p of a subset A of a

is ...

13 Let X be a B - space , and let K * be a bounded X - closed convex subset of X *

. Show that K * has a continuous tangent functional at each point of a dense

subset of its boundary . 14 DEFINITION . A point p of a subset A of a

**metric space**is ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero