## Linear Operators, Part 1 |

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Page 494

2 we conclude that T * maps the unit sphere of X * into a conditionally weakly

compact set of rca ( S ) , and therefore T * is a weakly

Theorem 4 . 8 this implies that T is a weakly

THEOREM .

2 we conclude that T * maps the unit sphere of X * into a conditionally weakly

compact set of rca ( S ) , and therefore T * is a weakly

**compact operator**. ByTheorem 4 . 8 this implies that T is a weakly

**compact operator**. Q . E . D . 4THEOREM .

Page 539

operator and its importance was clearly recognized by E . Schmidt [ 1 ] , to whom

the geometrical terminology in linear space theory is due . Compact and weakly

operator and its importance was clearly recognized by E . Schmidt [ 1 ] , to whom

the geometrical terminology in linear space theory is due . Compact and weakly

**compact operators**. The concept of a compact ( or completely continuous ) ...Page 540

Nelson Dunford, Jacob T. Schwartz. operator and the operator mapping into a

weakly complete space X . Weakly

treated by Sirvint [ 3 ] . A very incisive discussion of weakly

with ...

Nelson Dunford, Jacob T. Schwartz. operator and the operator mapping into a

weakly complete space X . Weakly

**compact operators**from C [ 0 , 1 ] to X weretreated by Sirvint [ 3 ] . A very incisive discussion of weakly

**compact operators**with ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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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 obtained 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