## Linear Operators, Part 1 |

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Page 37

If T : X + Y and U : Y + 3 are linear transformations , and X , Y , Z are linear spaces

over the same field Ø , the product UT ... If T is a

said to be a

...

If T : X + Y and U : Y + 3 are linear transformations , and X , Y , Z are linear spaces

over the same field Ø , the product UT ... If T is a

**linear operator**on X to X , it issaid to be a

**linear operator**in X. For such operators the symbol T2 is used for TT...

Page 486

The set of compact operators is closed in the uniform operator topology of B ( X ,

Y ) . ... Linear combinations of compact

and any product of a compact

...

The set of compact operators is closed in the uniform operator topology of B ( X ,

Y ) . ... Linear combinations of compact

**linear operators**are compact operators ,and any product of a compact

**linear operator**and a bounded**linear operator**is a...

Page 513

13 If U : Y * → X * is a linear mapping which is continuous with the y topology in Y

* and the X topology in 2 * , then there exists a bounded

such that T * = U. 14 Let T be a linear , but not necessarily continuous , mapping ...

13 If U : Y * → X * is a linear mapping which is continuous with the y topology in Y

* and the X topology in 2 * , then there exists a bounded

**linear operator**T : X + Ysuch that T * = U. 14 Let T be a linear , but not necessarily continuous , mapping ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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### Common terms and phrases

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