## Linear Operators: General theory |

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Page 104

It is customary to speak of the elements of

rather than sets of equivalent

shall write / instead of [/] and think of

...

It is customary to speak of the elements of

**F**(S, E, (i, X) as if they were**functions**rather than sets of equivalent

**functions**and this we shall ordinarily do. Thus, weshall write / instead of [/] and think of

**F**(S, E, /i. X) as the set of all**functions**on S to...

Page 182

j£

of Theorems 2 and 7 is called the Radon-Nikodym derivative of X with respect to

p and is often denoted by dkjdp. Thus dXjdp is defined /<-almost everywhere by ...

j£

**f**(s)p(ds) = j£ g(s)v(p, da) = ;.(£). Q.E.D. Note: The unique //-integrable**function**/of Theorems 2 and 7 is called the Radon-Nikodym derivative of X with respect to

p and is often denoted by dkjdp. Thus dXjdp is defined /<-almost everywhere by ...

Page 196

For each s in S, F(s) is an equivalence class of functions, any pair of whose

members coincide A-almost everywhere. If for each s we select a particular

T,ET,Z) ...

For each s in S, F(s) is an equivalence class of functions, any pair of whose

members coincide A-almost everywhere. If for each s we select a particular

**function f**(s, □) € F(s), the resulting**function f**(s, t) defined on (R,Er,q) = (S,Es,fi)x(T,ET,Z) ...

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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