## Linear Operators: General theory |

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Page 37

3 are linear transformations, and X, ?), 3 are linear spaces over the same field 0,

the product UT, defined by (UT)x = U(Tx), is a linear transformation which maps X

into 3- If T is a

3 are linear transformations, and X, ?), 3 are linear spaces over the same field 0,

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 be a**linear operator**in X. For ...Page 494

Conversely, let fj. be an X-valued measure defined on the Borel sets in S 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 fj. be an X-valued measure defined on the Borel sets in S 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 ...

Page 513

13 If U : 9)* -> X* is a linear mapping which is continuous with the 9) topology in

9J* and the X topology in X*, then there exists a bounded

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

...

13 If U : 9)* -> X* is a linear mapping which is continuous with the 9) topology in

9J* and the X topology in X*, then there exists a bounded

**linear operator**T : X ->9) such that T* = U. 14 Let T be a linear, but not necessarily continuous, mapping

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Common terms and phrases

a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set open set operator topology positive measure space Proc Proof properties proved real numbers Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero