## 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 , defined by ( UT ) x = U ( Tx ) , is a linear

transformation which maps X into 3. If T is 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 , defined by ( UT ) x = U ( Tx ) , is a linear

transformation which maps X into 3. If T is a

**linear operator**on X to X , it is said to...

Page 486

Linear combinations of compact

product of a compact

Linear combinations of compact

**linear operators**are compact operators , and anyproduct of a compact

**linear operator**and a bounded**linear operator**is a compact**linear operator**. Proof . The conclusions of Theorem 4 follow readily from the ...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