## Linear Operators: General theory |

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Page 37

If T : 36 -> 7) and U : '2) -» 3 are linear transformations, and 36, D, 3 are linear

spaces over the same field 0, the product UT, defined by (UT)x = U(Tx), is a linear

transformation which maps 36 into 3- If T is a

...

If T : 36 -> 7) and U : '2) -» 3 are linear transformations, and 36, D, 3 are linear

spaces over the same field 0, the product UT, defined by (UT)x = U(Tx), is a linear

transformation which maps 36 into 3- If T is a

**linear operator**on 36 to 36, it is said...

Page 490

We first consider operators with range in C(S) and then operators defined on C(S

). 1 Theorem. Let S be a compact Hausdorff space and let T be a bounded

We first consider operators with range in C(S) and then operators defined on C(S

). 1 Theorem. Let S be a compact Hausdorff space and let T be a bounded

**linear****operator**from a B-space X into C(S). Then there exists a mapping x : S -> X* ...Page 494

Conversely, let fi be an X-valued measure defined on the Borel sets in 5 with the

property that x*fi is in rca(S) for every x* in X*. It is clear that the operator T,

defined by (b), is a bounded

given by ...

Conversely, let fi be an X-valued measure defined on the Borel sets in 5 with the

property that x*fi is in rca(S) for every x* in X*. It is clear that the operator T,

defined by (b), is a bounded

**linear operator**on C(S) to X whose adjoint T* isgiven by ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero