## Linear Operators: General theory |

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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 Z. 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 Z. If T is a

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

Page 494

It is clear that the operator T , defined by ( b ) , is a bounded

S ) to X whose adjoint T * is given by ( d ) . From IV.10.2 we conclude that T *

maps the unit sphere of X * into a conditionally weakly compact set of rca ( S ) ,

and ...

It is clear that the operator T , defined by ( b ) , is a bounded

**linear operator**on C (S ) to X whose adjoint T * is given by ( d ) . From IV.10.2 we conclude that T *

maps the unit sphere of X * into a conditionally weakly compact set of rca ( S ) ,

and ...

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 X * , 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 X * , 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

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero