## Linear Operators: General theory |

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Page 37

If T : X + Y and U : Y → Z 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

said ...

If T : X + Y and U : Y → Z 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 issaid ...

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 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 ) ...

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 ) ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

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