## Linear Operators: General theory |

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Page 410

1 Definition. A

The following lemma is an obvious consequence of Definition 1. 2 Lemma. The

intersection of an arbitrary family of

1 Definition. A

**set**K Q X is**convex**ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K.The following lemma is an obvious consequence of Definition 1. 2 Lemma. The

intersection of an arbitrary family of

**convex**subsets of the linear space X is**convex**.Page 461

and an arbitrary

Theorem 2 . 8 ) . He also proved that a

topology of the normed linear space X , can be separated from an arbitrary

closed ...

and an arbitrary

**convex set**is possible , provided they are disjoint ( compareTheorem 2 . 8 ) . He also proved that a

**convex set**K which is compact in the X *topology of the normed linear space X , can be separated from an arbitrary

closed ...

Page 838

space of, definition, IV.2.24 (242) properties, IV.15 Annihilator of a set, II.4.17 (72)

Arzela theorem, on continuity of limit ... IV.4.12 (252) Basic separation theorem

concerning

space of, definition, IV.2.24 (242) properties, IV.15 Annihilator of a set, II.4.17 (72)

Arzela theorem, on continuity of limit ... IV.4.12 (252) Basic separation theorem

concerning

**convex sets**, V.I. 12 (412) Bernstein theorem, concerning cardinal ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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