## Linear Operators: General theory |

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Page 37

J -*□ 3 are linear transformations, and X, *J), 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

J -*□ 3 are linear transformations, and X, *J), 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**...Page 486

The set of compact operators is closed in the uniform operator topology of B(7i,

?)) ... Linear combinations of compact

and any product of a compact

...

The set of compact operators is closed in the uniform operator topology of B(7i,

?)) ... Linear combinations of compact

**linear operators**are compact operators,and any product of a compact

**linear operator**and a bounded**linear operator**is a...

Page 494

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

to 32 whose adjoint T* is given by (d). From IV. 10.2 we conclude that T* maps

the unit sphere of 36* 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 32 whose adjoint T* is given by (d). From IV. 10.2 we conclude that T* maps

the unit sphere of 36* into a conditionally weakly compact set of rca(S), and ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional linear map linear operator linear topological space LP(S measurable functions 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 Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact